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Chapter 16 Market Evaluation through Transition Probabilities
CHAPTER 16
Market Evaluation through Transition Probabilities Several approaches have thus far been used in the study of market trends and market potential. Some of these approaches are more or less mathematical, while others could easily be described as “largely subjective.” In either case the objective of market research is to detect, as accurately as possible, the wishes of the human population that constitutes the “market.” In turn, this means finding out the feelings of the buyer, and, if the buyer is a stochastic element in himself, his feelings are twice as stochastic. T h e most obvious reason why mathematical analysis of the microevent that takes place in a market has proved to be less dependable than market analysts were hoping for can be found in the fact that the singular psychological effect escapes mathematical thinking. A model of behavior becomes even more probabilistic when one wishes to integrate into it social and economic aspects-an area in which we have had thus far very few successes but a substantial variety of failures. This fact alone calls for more research. For this reason, in the present chapter we will advance the hypothesis that if the micro-event defies mathematical analysis, at the present state of the art at least, the macro-event may be more amenable to the analytical approach. In studying the market behavior of human populations, we will use analogies from the behavior of colonies of insects. Our mathematical tool will be the matrices of transition probabilities. I t is understood that this is not an exact model but an attempt to examine how Markoff chains can be applied to the study of macro-events, to obtain general market tendencies and drives. It is understood that much more research is necessary if we are going to obtain transitional media capable of being used in “far-out planning.” T h e whole field is yet in its infancy. Furthermore, the approach discussed in this chapter should not be considered an exclusive one. In Chapter 1 I we have paid due consideration to mathematical studies oriented toward the discovery of rules that guide the structural behavior of a market. 295
296
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Application of Markoff Chains in the Automotive Industry by Car Model T h e information in Table I1 indicates the relative position of the TABLE I COMPARATIVE POSITION OF AUTOMAKES BY YEAR,BASEDON NUMBER OF CARSSOLD
I947 ~
1
2 3 4 5 6 7 8 9 10
Chev Ford Plyrn Buick Dodge Pant Olds Merc Nash Stud
1948
49
51
50
~-
~
~
Chev Chev Chev Ford Ford Ford Plym Plyrn Plyrn Buick Buick Buick Pont Pont Pont Dodge Dodge Olds Olds Olds Merc Stud Stud Dodge hlerc Merc Stud Hudson Hudson Nash
~-
Chev Ford Plym Buick Pont Dodge Olds Merc Stud Chrys
52
53
54
55
Chev Ford Plyrn Buick Pont Dodge Olds Merc Stud Nash
56
~
~~
Chev Ford Plyrn Buick Pont Olds Merc Dodge Stud Chrys
Chev Ford Buick Olds Plyrn Pont Merc Dodge Cad Chrys
Chev Ford Buick Plyrn Olds Pont Merc Dodge Chrys Cad
Chev Ford Buick Plyrn Olds Pont Merc Dodge Cad DeSoto
various American automobiles, based on the number of cars sold each year. From these data we try to project on changes in position which may occur. We start by analyzing this information to obtain a group of trajectories showing the transitions that take place from “relative position” to “relative position” in the automobile market. T h e trajectories are as in Table 11, where the given numbers refer to the relative position. TABLE I1 Chevrolet Ford Plymouth hick Dodge Pontiac Oldsmobile Mercury Nash Studebaker Hudson Cadillac Chrysler De Soto
I 2 3 4 5 6 7 8 9 10
>I0 ;I0 >I0 >I0
1
2 3 4 6 5 7 9 >I0 8 10
>I0 >I0 >I0
1
;I0
2 3 4 6 5 7 9 8
10
>I0 >I0 >I0
1
2 3 4 8 5 6 7 10
9 >I0 >I0
>I0 >I0
From Ward’s 1957 Automotive Yearbook.
I 2 3 4 6 5 7 8
>I0 9 >I0 >I0 >10
I 2 3 4 6 5 7 8 10 9
>I0 >I0 >I0 110 > I 0
,I0
I 2 3 4 8 5 6 7
9 ;I0 >I0
10
;I0
I 2 5 3 8 6 4 7 /I0 >I0
>to
>9 10 /I0
1
2 4 3 8 6 5 7 >I0 >I0 >I0
10
9 >I0
1
2 4 3 8 6 5 7
,I0 >I0
>I0 9 >I0
10
MARKOFF CHAINS BY CAR MANUFACTURER
297
A trajectory is easily obtained for each car under consideration simply by assigning a numerical value to its relative position, as for instance 1 for first, 2 for second, ... and > 10 for a car which ceases to appear in the table of the first ten places. Then, by counting the frequency of each individual change, for instance from 6 to 5, the probability of occurrence of that change can be established. These probabilities would then need to be assembled into a matrix, and, if the probabilities found for each transition are unchanged over long periods of time, the matrix may be considered to define a Markoff chain. From this matrix, we can determine the tendencies of the system. T h e broad conclusions which can be derived from the relative positions in Table 11 are as follows: T h e cars on top of the matrix will stay on top, the cars on the bottom have a good chance of dropping off the list, and the cars in the middle will continue to interchange positions. Based on these data, we will use a matrix of transition probabilities in studying the change in status effective within the “make-market” system, and will attempt to forecast its future status. However, it should be noted that this effort is based on the assumption that the data of Table I1 lead to a Markoff chain. I n certain cases, the validity of this assumption may be questionable. Conclusions drawn from such studies must be accepted within the limitations of explicitely stated considerations. Markoff Chains by Car Manufacturer Knowing from historical information the relative gains and losses for each of the major automobile manufacturing companies, it was possible to compute for every year the choice of the automobile-buying population among alternative makes. This population will hereafter be referred to as the “market.” For an example on how the data were treated we consider the year 1954. During that year, the following manufacturers showed an increase in the market:
( I ) Ford Motor Company (5.68 yo), (2) General Motors (5.63 yo), (3) Overseas makes (0.09 yo).2 Manufacturers showing a decrease during this same year, 1954, were Chrysler Motors Company (7.41 yo) and all other American makes Taken in the commulative.
298
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
For manufacturers who showed a gain in the market, the portion of the total gain that each controlled can be expressed as a ratio: 5.68 . Ford Motors, 11.40 ’
(3.99
General Motors,
5.63 . 11.40 ’
Overseas makes,
0.09 11.40
Hence, the portion of the market which Chrysler surrendered to each of the manufacturers “in the increase” is equal to Ford Motors,
5.68 11.40
3.99%
=
1.99% ;
General Motors,
-X 3.99%
=
1.97% ;
Overseas makes,
___ X
X
5.63
1 1.40
0.09 1 1.40
3.99%
= 0.030/0
.
Similar calculations can be made for the remaining years under consideration, and such data can be tabulated as tables which show the transitions of the market percentages from one year to the next. Table I11 presents historical information on new passenger car registration TABLE I11 NEWPASSENGER CARREGISTRATION BY THE MAJOR AUTOPRODUCERS, YEARSI95 I - I 957 Auto manufacturer
Number of units
1951
1952
1953
1954
1955
1956
1957
Chrysler I I03 330 884667 I I65 357 713 347 1206195 922043 I096359 Motors FordMotors I121464 947474 1443153 1706617 I980736 1694108 1818169 GeneralMotors 2167 713 I735 694 2586 697 2006595 3639 120 3024286 2683 365 All other U.S. models 646758 561 260 514821 282520 292199 223 769 177622 Foreign models 20 828 29 299 28 961 25 385 51 658 91 042 206 827 Totals
5060093 4158 394 5738989 5535464 7 169908 5955248 5982342
Also lumped together into one classification.
299
MARKOFF CHAINS BY CAR MANUFACTURER
for years 1951 to 1957; Table IV gives the same information in percent of total new car registration; and Table V presents the percentage of gain or loss of new car registration for the major automobile manufacturers. T h e latter are based on the percentages in Table IV, taking 1951 as the starting year. TABLE IV
SAME INFORMATION AS IN TABLE 111, EXPRESSED I N yo OF TOTAL NEWCARREGISTRATIONS
% of the total
Auto ~~
manufacturer Chrysler Motors Ford Motors General Motors All other U.S. models Foreign models Totals
1951
1952
1953
1954
1955
1956
1957
21.80 22.16 42.83 12.00 0.41
21.27 22.78 41.74 13.51 0.70
20.31 25.15 45.07 8.97
0.50
12.90 30.83 50.70 4.98 0.59
16.82 27.63 50.75 3.98 0.82
15.84 28.45 50.78 3.64 1.65
18.34 30.39 44.85 2.96 3.46
100.00
100.00
100.00
100.00
100.00
100.00
100.00
TABLE V PERCENT OF GAINOR LOSS OF THE NEW CAR %GISTRATIONS FOR MAJOR AUTOMOBILE MANUFACTURERS (BASED ON PERCENTAGES IN TABLE I1 WITH 1951 AS THE STARTING YEAR) Auto manufacturer Chrysler Motors Ford Motors General Motors All other U S . models Foreign models Total increase
Percent gain or loss with respect to preceding year 1951
1952
1953
1954
1955
1956
1957
0
0
-0.53 0.62 -1.09 0.71 0.29
-0.96 2.37 3.33 4.54 -4.20
-7.41 5.68 5.63 -3.99 0.09
3.92 -3.20 0.05 -1.00 0.23
-1.34 0.82 0.03 4.34 0.83
2.86 1.94 -5.93 -0.68 1.81
0
1.62
5.70
11.40
4.20
1.68
6.61
0 0
0
Tables VI through XI offer information on the market transition for the major automobile manufacturers, expressed in percentage of the total market, on a year-to-year basis. I n Table VI the market transition is from 1951 to 1952, in Table VII from 1952 to 1953, and so on to Table XI whose data correspond to the market transition from 1956 to 1957.
300
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
TABLE VI MARKET TRANSITIONS (EXPRESSED I N PERCENTAGE OF TOTAL MARKET) FROM 195 I
C F G A 0 1951 Totals
TO
1952"
C
F
G
A
0
1952 Totals
21.27 0.20 0.00 0.24 0.09
0.00 22. I6 0.00 0.00 00:00
0.00 0.42 41.75 00.47 0.20
0.00 0.00 0.00 12.00 0.00
0.00 0.00 0.00 0.00 0.41
21.27 22.78 41.74 13.51 0.41
21.80
22. I6
42.83
12.80
0.41
100.OO~o
Key to symbols in this and following tables: C, Chrysler Motors; F, Ford Motors; G, General Motors; A, all other American makes; 0, Overseas (foreign) makes. TABLE VII MARKET TRANSITIONS FROM 1952
C F
G A
0
1952 Totals
C
F
20.31 0.40 0.56 0.00 0.00
0.00 22.78 0.00 0.00 0.00
21.27
22.78
TO
1953
A
0
1953 Totals
0.00 0.00 41.74 0.00 0.00
0.00 1.89 2.65 8.97 0.00
0.00 0.08 0.12 0.00 0.50
20.31 25.15 45.07 8.97 0.50
41.74
13.51
0.70
100.00yo
0
1954 Totals 12.90 30.83 50.70 4.98 0.59
G
TABLE VIII MARKET TRANSITIONS FROM 1953
C F G
A
0
1953 Totals
TO
1954
C
F
G
A
12.90 3.69 3.66 0.00 0.06
0.00 25.15 0.00 0.00 0.00
0.00 0.00 45.07 0.00 0.00
0.00 1.99 1.97 4.98 0.03
0 0 0 0 0.50
20.31
25.15
45.07
8.97
0.50
100.OOo/o
30 1
MARKOFF CHAINS BY CAR MANUFACTURER
TABLE IX MARKET TRANSITIONS FROM 1954
C F
G A 0
1954 Totals
TO
1955
C
F
G
A
0
1955 Totals
12.90 0.00 0.00 0.00 0.00
2.99 27.63 0.04 0.00 0.17
0.00 0.00 50.70 0.00 0.00
0.93 0.00 0.01 3.98 0.06
0.00 0.00 0.00 0.00 0.59
16.82 27.63 50.75 3..98 Oi82
1290
30.83
50.70
4.98
0.59
100.OOo/o
TABLE X MARKET TRANSITIONS FROM 1955
C F G
A 0 1955 Totals
TO
1956
C
F
G
A
0
1956 Totals
15.48 0.66 0.02 0.00 0.66
0.00 27.63 0.00 0.00 0.00
0.00
0.00 50.75 0.00 0.00
0.00 0.16 0.01 3.64 0.17
0.00 0.00 0.00 0.00 0.82
15.48 28.45 50.78 3.64 1.65
16.82
27.63
50.75
3.98
0.82
lOO.OOO/o
A
0
1957 Totals
TABLE XI MARKET TRANSITIONS FROM 1956
C F G A 0
1956 Totals
TO
1957
C
F
G
15.48 0.00 0.00 0.00 0.00
0.00 28.45 0.00 0.00 0.00
2.57 1.74 44.85 0.00 1.62
0.29 0.20 0.00 2.96 0.19
0.00 0.00 0.00 0.00 1.65
18.33 30.39 44.85 2.96 3.47
15.48
28.45
50.78
3.64
1.65
lOO.OOO/o
302
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
On the hypothesis that the available sample of six years was representative enough of the corresponding population of transition,the experimenters tested various approaches to the continuation of the work. One of the approaches involved the use of the mode or the median as the “representative” value for a transition cell. Having obtained unsatisfactory results they tested certain other “values” derived from the available data. They finally decided to experiment with the sum-total of the transition values for the same cell. By summing up the corresponding values in Tables VI through XI, the data in Table XI1 were obtained. TABLE XI1 SUMMATION OF TABLES VI
TO
XI
C
F
G
A
A 0
98.34 4.95 4.24 0.23 0.81
2.98 153.80 0.04 0.00 0.18
2.56 2.16 284.89 0.48 1.83
1.22 4.23 2.68 34.33 0.45
0.00 0.08 0.12 0.00 4.47
Totals
108.57
157.00
291.92
42.91
4.67
C
F G
0
If we let Table XI1 represent the transition from the “present year” T to the next year T 1, the summation of each of the columns gives a number representing the portion of the market that one automobile manufacturer had in relation to his four competitors. Taking year T as the reference for future years, T 1, T 2, T n, the columns of the Table in question can be expressed in a percent form. T h e transformated matrix is given in Table XIII.
+
+
+
+
TABLE XI11
TRANSFORMATION MATRIX FROM YEART C
F
G
TO
YEART
+I
A
0 .
C F G A 0
0.905 0.046 0.039 0.002 0.008
0.019 0.980 0.000 O.OO0 0.001
0.009 0.007 0.976 0.002 0.006
0.028 0.099 0.062 0.801 0.010
0.000 0.017 0.026 0.000 0.957
Totals
1 .Ooo
1 .om
1 .000
1.Ooo
1 .ooo
-
MARKOFF CHAINS BY CAR MANUFACTURER
303
T h e data in Table XI11 can also be represented in a graphical form, as in Fig. 1. Arrows indicate the direction of the change, and the corresponding percentages are duly noted.
Fic. 1
Assuming that the information presented in Table XI11 corresponds to a state of equilibrium,* from these same data we can derive a set of transition equations. For instance, let C’, F’,G‘, A’, and 0’ represent the components of a vector resulting from the previous state (C, F , G, A , 0). T h e equations derived from the information in the transition matrix, expressing the relation between the components of the two vector states, would be as follows: C’ = 0.905C F’ = 0.046C G’ = 0.039C A’ = 0.002C 0‘ = 0.008C
+ 0.019F + 0.009G + 0.028A, + 0.980F + 0.007G + 0.099A + 0.0170, + 0.9766 + 0.062A + 0.0260, + 0.002G + 0.801A, + 0.001F + 0.006G + 0.010A + 0.9570;
or the resultant vector X, is derived by multiplying the transition matrix [MI by the vector X , which is the state C, F,G, A , 0 of the system. A supposition,
16.
304
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
If X , represents the vector after n transformations, this could be expressed as X , = [MIX,-, . Based on the same suppostion, the experimenters derived simulated data regarding the percent share of the market for the competing automobile manufacturers. They then compared this information against the available historical data. T h e results of the comparison are presented in Figs. 2 through 6. Three simulated data curves are plotted in each
22 1951
1952
1953
I954
1955
I956
1957
Year FIG.2. New Ford Motors cars registered in C7.S. during years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for I-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
figure. Each corresponds to a different cycling procedure. Namely, the hypothesis was made that if other things are kept equal and the cycle of transition is varied the resulting simulated curve may approximate more closely the historical data, which in every case are represented by curve 1. Three alternative cycles have been considered, corresponding to
305
MARKOFF CHAINS BY CAR MANUFACTURER
curves 2, 3, and 4. Curve 2 is based on simulated data for a one-year cycle. For this curve the matrix of transition probabilities given in Table XIV was used. Curve 3 corresponds to simulated data for a two-year cycle, while curve 4 reflects the outcome of a simulation involving a three-year cycle. This reference to curves I , 2, 3 , and 4 is
&
1
20
In E
0 ..a-
e
c
.-In 0 2
18
\
\
L
0
u B
0)
0
16
c 0 c
+ 0 c
-
.-0
bQ
14
In Q In L
r'
2,
\
12
0
\
\ ' \
10
1951
I
1952
I
1953
I
I
I
1454
1955
1956
Year
1 1957
FIG. 3. New Chrysler Motors cars registered in U.S. during years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
valid for all figures under consideration. T h e rationale for the twoyear and three-year cycle hypotheses was that the American market is getting into the habit of changing the low-priced cars on a two-to-three years basis. T h e resulting graphical plot of the simulated data indicates that the year-to-year basis is the closest one to the actual. In Fig. 2 (Ford Motor
306
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Company), curve 2 indicates with considerable accuracy the trend of the company with respect to percent share of the market. Curve 3 gives, generally, an overestimate and only once matches the actual data, while Ford cars sales were pick. Finally, curve 4 grossly overestimates market trends.
-$ Y
50-
v)
c
0 .-c
i
‘548 ?!
’
e
L
3
Q
-
-
46
0
c
0
c
w-
0
c 0 .-
z
4-
44
Q
e0
c
0
42
c
Q,
c3 40
I
1951
1
I
I
I
I
1952
1953
1954
1955
1956
1 1957
Year FIG.4. New General Motors cars registered in U.S. during the years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
I n the case of Chrysler Motors, curve 3 presents the best approximation of the trend in the company’s percent share of the market. T h e approach represented by curve 2 seems to be “optimistic,” while that of curve 4 deviates considerably from the actual data. Comparing the results for Ford and Chrysler of approach 4, we see that in both cases this three-year cycling has a tendency to go to extremes; when the
307
MARKOFF CHAINS BY CAR MANUFACTURER
percent share of the market is ascending it overestimates the gains, when it is descending it overestimates the losses. It should be noticed, however, that, as considered here, the approach of curve 4 is based on a smaller sample than either that of curve 2 or 3. I n the case of the General Motors, all three approaches failed to
I--- , 1952
1953
1954
1955
1956
1957
Year FIG. 5. All other new American-made cars ,registered in the U.S. during the years 1951-1957 (all those not produced by Ford, Chrysler, or General Motors). Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles. reflert t h e l a r o e inrreace in
the
r n m n a n v ’ c charp nf t h e market fnr
the
years 1954, 1955, and 1956. Curve 2 still indicates the trend in percent share over the whole six-year span without the above-mentioned gains, while curves 3 and 4 present an off-phase increase in share, and they both are sensitive in the corresponding decrease during 1957. Figure 5 presents actual and simulated data for the other American
308
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
makes. While all three simulated curves reflect the trend in the market, curve 2 is the closest to the real data. With the overseas makes, as seen in Fig. 6, curve 2 gives the closest approximation. On the basis of the foregoing, it can reasonably be concluded that Markoff chains may eventually present good planning possibilities, but more research is necessary, both to establish its areas of best potential and to develop the sensitivity of the method.
1951
1952
1953
1954
1955
l95b
1957
Year FIG. 6. New overseas (foreign) made cars registered in the U.S. during the years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for2-year cycles, and curve 4 on simulated data for 3-year cycles.
A Used-Car Model For another application of Markoff chains, an evaluation was made of the changes of automobile makes that take place when used cars are traded in for new cars. T h e obtained information is mostly from estimates made by car dealers in the Washington, D. C. area.
309
A USED-CAR MODEL
TABLE XIV CARTRADE-IN PERCENTAGES Car traded in
New car buy Chevrolet Ford Plymouth Others
Chevrolet
Ford
Plymouth
Others
0.85 0.06 0.04 0.05
0.09 0.82 0.05 0.04
0.03 0.03 0.90 0.04
0.10 0.10 0.10 0.70
T h e information in question is presented in Table XIV in the 'form
of a matrix of transition probabilities. If we assume that 100 cars of
each make: Chevrolet, Ford, Plymouth, and Others (lumped together) are traded in each year, the distribution of cars at the end of each year would be as shown in Table XV. T h e results indicate that if the trade-in percentages do not change, and if the impact of first-time buyers is ignored, Chevrolet will have greatly improved its position, and the Other makes will have very reduced sales. These calculations are made on the basis of suppositions advanced by car dealers and reflect their way of thinking and of projecting into the future of their trade. The model was developed as a minor research project in 1957, and, since then, market evolution has generally tended to support the obtained results. TABLE XV DISTRIBUTION OF CARSAFTER 1 Years
Chevrolet
TO
Ford
15 YEARS Plymouth
Others
100
100 83 73 67 62 59 56 54 53 52 52 53 54 55 55 55
~~
0
100
1
107
2 3 4 5 6 7 8 9
ill
10 11
12 13 14 15
1 I3 115 1 I7
118 1 I9 119 118 1 I7
116 115 1 I4 I I4 1 I4
100 101 101 100 100
99 98 97 95 95 94 93 92 91 91 91
109 1 I5 120 123 125 I28 130 133 135 137 138 139 I40 I40 140
3 10
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Studying the Impact of the ABC Car on the American Economy Over a number of years, considerable discussion has taken place in the United States on the rationale of the yearly change in model by all major American car manufacturers. T h e manufacturers have strongly maintained that the yearly change in model is the lifeblood of the growth in the automobile industry, and, therefore, both in the long and in the short run it is to the benefit of the nation. T o document their reports, car manufacturers have insisted that without the yearly change in design the automobile market would limit itself to the replacement of scrapped cars, or below the fourmillion-unit mark. In contrast, automobile sales for the last eight years varied in the range of five to seven and a half million units due to the artificial obsolescence or, in different terms, to the yearly change in model. On the other side of the fence, some economists argue that these yearly model changes with the corresponding retooling etc. are a kind of waste and that the American economy can do as well without them. T o test the validity of these arguments, we will assume that a “new” automobile company would design, produce, and market the American Basic Car, hereafter referred to as ABC, with a guaranteed no-change in model for a period of twelve years.5 T h e engineering dependability of the car is also set at the same twelve-year level. By this token, a buyer of the ABC would have no inducement, either because of an artificial or real obsolescence, to trade in this car before the 12-year period is over. T h e objective is to study the effect of this policy on the American market. Say that the XBC car is built in the size range of the Chevrolet-FordPlymouth lines with an economy engine. Say also, that because of its novelty the first year of being introduced into the market it captures somewhat over 54 yo of the total car sales, dropping thereafter to 50 yo. By the same token, a certain car market of (say) 5 500 000 cars would go up to 6 OOO 000 due to ABC’s advent. Finally, it is assumed that the “other” cars are traded in every three years, as they used to up to now and that, in every trade-in model, ABC takes the 50 yo of the buyers. Some of the quantitative aspects of these assumptions may be regarded as unrealistic, yet the impact the compact cars had in the year of their introduction could help document that this is not so bad an estimate.
‘
To ease the questions which will automatically arise in regard to a marketing and servicing network, it is acceptable to assume that ABC represents the “Basic Cars” of all established manufacturers lumped together. The same twelve-year policy is assumed to prevail.
STUDYING THE IMPACT OF ABC CAR ON AMERICAN ECONOMY
311
A second tentative statement must also be made. I t concerns a static car market. With it, the result will be a major upheaval in the American economy. T h e upheaval would be reflected in the substantial shrinkage of the labor demand, a decline in the metals and other materials industries, a substantial reduction in numbers of dealers, advertisers, and auto service companies, and a decline in auto-parts manufacturing concerns. These changes would have an adverse affect on the purchasing power of the American public with an accompanying drastic shrinkage in all phases of the economy. We will assume that the “present” market is divided as follows: Northern Company Western Company Eastern Company Independents
32% 29% 25% 14%
T h e owners of Big Three’s cars, in 85 yo of all cases, trade for the same car. T h e remaining 15 yo go about even to the two other makes or to the independents. Owners of independent products stick, about 91 yo,with their makes; the remaining 9 yo distribute themselves about evenly among the big three makes. From these data, the following matrix of annual transformation can be developed:
Northern Western Eastern Indep.
Northern (N) 0.85N O.05N 0.05N O.OSN
Western (W) 0.05W 0.85W 0.05W 0.05W
Eastern (E) O.05E 0.05E 0.85E 0.05E
Indep. (D) 0.03D 0.03D 0.03D 0.91D
I n the year “zero” based on the given percentages of the market, Northern manufactures Western manufactures Eastern manufactures Independents manufacturers Total
I 760 000 Cars I 595 000 Cars I 375 000 Cars 770 000 Cars 5 500 000 Cars
In the ensuing years the market changes as follows: Year I
Northern Western Eastern Indep.
1 667 600 1 535 000
Year 2 1 588 100
I 399 000 937 200
I 484 700 1 346 100 1081 100
5 500 000
5 500 000
3 12
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
T h e big three are losing ground because of the hypothesis made that they command a lower customer loyalty than the independents. Now say that into this market, at year “one,” the new car is introduced by the ABC Company. This car guarantees the stated twelve-year life and no model change during the same period. We made reference to a market of 6 000 000 units the first year and some 54 yo of the share would be gained by the ABC car the first year. For the subsequent years the matrix of transformation is Northern Western Eastern Indep. ABC
Northern 0.425N 0.025N 0.025N 0.025N 0.500N
Western 0.005W 0.425W 0.025W 0.025W 0.500W
Eastern 0.025E 0.025E 0.425E 0.025E 0.500E
Indep. 0.015D 0.015D 0.015D 0.455D 0.500D
After the first year, the hypothetical American market will continue for two more years as shown in the matrix for the pre-ABC period. TABLE XVI DISTRIBUTION OF CARSAT
THE
END OF EACH.YEAR
~~
Year
Company Northern Western Eastern Indep. ABC Total
833 800 767 800 679 800 468 600 3 250 000
1
2 794 050 742 350 673 060 540 550 2 750 000
3 761 500 720 000 665 500 620 000 2 750 000
4 368 500 350 300 327 800 328 400 1 375 000
6 000 000
5 500 000
5 500 000
2 750 000
Northern Western Eastern Indep. ABC Total
5 357 000 342 400 324 400 351 200 1375 000 2 750 000
6 347 300 335 600 321 300 370 800 1375 000 2 750 000
7 169 600 164 900 159 200 193 800 687 500 1 375 000
8 166 200 I62 400 157 900 201 000 687 500 1 375 000
9 163 300 160 300 156 700 207 200 687 500
10 80 400 79 200 77 800 106 300 343 750
11 79 350 78 450 77 200 108 750 343 750
12 78 700 77 700 76 700 110 750 343 750
1375 000
687 500
687 500
687 500
Northern Western Eastern Indep. ABC Total
STUDYING THE IMPACT OF ABC CAR ON AMERICAN ECONOMY
313
I n the fourth year, however, the ABC owners will not trade their cars as they consider them good, engineering and designwise, for 12 years. This behavior will repeat itself to the twelfth year. T h e effect of the super-durable car can be seen in Table XVI. T h e “old line” companies, which before the advent of the ABC had manufactured 5 500 000 cars per year, twelve years later are down to 343 750 cars. T h e ABC Company, while always maintaining at least 50% of the market, drops from 3 250 000 cars in year “one” to 343 750 cars in the 12th year. T h e result, therefore, of this super-durable inovation would be a virtual collapse of the industry. Though this is just a hypothesis and the chosen percentages, such as 50%, might be considered as an exaggeration, it is nevertheless true that the same result would have been obtained with small, “more reasonable” percentages] the magnitude of this outcome becoming essentially a matter of time.
Market Evaluation through Transition Probabilities Several approaches have thus far been used in the study of market trends and market potential. Some of these approaches are more or less mathematical, while others could easily be described as “largely subjective.” In either case the objective of market research is to detect, as accurately as possible, the wishes of the human population that constitutes the “market.” In turn, this means finding out the feelings of the buyer, and, if the buyer is a stochastic element in himself, his feelings are twice as stochastic. T h e most obvious reason why mathematical analysis of the microevent that takes place in a market has proved to be less dependable than market analysts were hoping for can be found in the fact that the singular psychological effect escapes mathematical thinking. A model of behavior becomes even more probabilistic when one wishes to integrate into it social and economic aspects-an area in which we have had thus far very few successes but a substantial variety of failures. This fact alone calls for more research. For this reason, in the present chapter we will advance the hypothesis that if the micro-event defies mathematical analysis, at the present state of the art at least, the macro-event may be more amenable to the analytical approach. In studying the market behavior of human populations, we will use analogies from the behavior of colonies of insects. Our mathematical tool will be the matrices of transition probabilities. I t is understood that this is not an exact model but an attempt to examine how Markoff chains can be applied to the study of macro-events, to obtain general market tendencies and drives. It is understood that much more research is necessary if we are going to obtain transitional media capable of being used in “far-out planning.” T h e whole field is yet in its infancy. Furthermore, the approach discussed in this chapter should not be considered an exclusive one. In Chapter 1 I we have paid due consideration to mathematical studies oriented toward the discovery of rules that guide the structural behavior of a market. 295
296
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Application of Markoff Chains in the Automotive Industry by Car Model T h e information in Table I1 indicates the relative position of the TABLE I COMPARATIVE POSITION OF AUTOMAKES BY YEAR,BASEDON NUMBER OF CARSSOLD
I947 ~
1
2 3 4 5 6 7 8 9 10
Chev Ford Plyrn Buick Dodge Pant Olds Merc Nash Stud
1948
49
51
50
~-
~
~
Chev Chev Chev Ford Ford Ford Plym Plyrn Plyrn Buick Buick Buick Pont Pont Pont Dodge Dodge Olds Olds Olds Merc Stud Stud Dodge hlerc Merc Stud Hudson Hudson Nash
~-
Chev Ford Plym Buick Pont Dodge Olds Merc Stud Chrys
52
53
54
55
Chev Ford Plyrn Buick Pont Dodge Olds Merc Stud Nash
56
~
~~
Chev Ford Plyrn Buick Pont Olds Merc Dodge Stud Chrys
Chev Ford Buick Olds Plyrn Pont Merc Dodge Cad Chrys
Chev Ford Buick Plyrn Olds Pont Merc Dodge Chrys Cad
Chev Ford Buick Plyrn Olds Pont Merc Dodge Cad DeSoto
various American automobiles, based on the number of cars sold each year. From these data we try to project on changes in position which may occur. We start by analyzing this information to obtain a group of trajectories showing the transitions that take place from “relative position” to “relative position” in the automobile market. T h e trajectories are as in Table 11, where the given numbers refer to the relative position. TABLE I1 Chevrolet Ford Plymouth hick Dodge Pontiac Oldsmobile Mercury Nash Studebaker Hudson Cadillac Chrysler De Soto
I 2 3 4 5 6 7 8 9 10
>I0 ;I0 >I0 >I0
1
2 3 4 6 5 7 9 >I0 8 10
>I0 >I0 >I0
1
;I0
2 3 4 6 5 7 9 8
10
>I0 >I0 >I0
1
2 3 4 8 5 6 7 10
9 >I0 >I0
>I0 >I0
From Ward’s 1957 Automotive Yearbook.
I 2 3 4 6 5 7 8
>I0 9 >I0 >I0 >10
I 2 3 4 6 5 7 8 10 9
>I0 >I0 >I0 110 > I 0
,I0
I 2 3 4 8 5 6 7
9 ;I0 >I0
10
;I0
I 2 5 3 8 6 4 7 /I0 >I0
>to
>9 10 /I0
1
2 4 3 8 6 5 7 >I0 >I0 >I0
10
9 >I0
1
2 4 3 8 6 5 7
,I0 >I0
>I0 9 >I0
10
MARKOFF CHAINS BY CAR MANUFACTURER
297
A trajectory is easily obtained for each car under consideration simply by assigning a numerical value to its relative position, as for instance 1 for first, 2 for second, ... and > 10 for a car which ceases to appear in the table of the first ten places. Then, by counting the frequency of each individual change, for instance from 6 to 5, the probability of occurrence of that change can be established. These probabilities would then need to be assembled into a matrix, and, if the probabilities found for each transition are unchanged over long periods of time, the matrix may be considered to define a Markoff chain. From this matrix, we can determine the tendencies of the system. T h e broad conclusions which can be derived from the relative positions in Table 11 are as follows: T h e cars on top of the matrix will stay on top, the cars on the bottom have a good chance of dropping off the list, and the cars in the middle will continue to interchange positions. Based on these data, we will use a matrix of transition probabilities in studying the change in status effective within the “make-market” system, and will attempt to forecast its future status. However, it should be noted that this effort is based on the assumption that the data of Table I1 lead to a Markoff chain. I n certain cases, the validity of this assumption may be questionable. Conclusions drawn from such studies must be accepted within the limitations of explicitely stated considerations. Markoff Chains by Car Manufacturer Knowing from historical information the relative gains and losses for each of the major automobile manufacturing companies, it was possible to compute for every year the choice of the automobile-buying population among alternative makes. This population will hereafter be referred to as the “market.” For an example on how the data were treated we consider the year 1954. During that year, the following manufacturers showed an increase in the market:
( I ) Ford Motor Company (5.68 yo), (2) General Motors (5.63 yo), (3) Overseas makes (0.09 yo).2 Manufacturers showing a decrease during this same year, 1954, were Chrysler Motors Company (7.41 yo) and all other American makes Taken in the commulative.
298
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
For manufacturers who showed a gain in the market, the portion of the total gain that each controlled can be expressed as a ratio: 5.68 . Ford Motors, 11.40 ’
(3.99
General Motors,
5.63 . 11.40 ’
Overseas makes,
0.09 11.40
Hence, the portion of the market which Chrysler surrendered to each of the manufacturers “in the increase” is equal to Ford Motors,
5.68 11.40
3.99%
=
1.99% ;
General Motors,
-X 3.99%
=
1.97% ;
Overseas makes,
___ X
X
5.63
1 1.40
0.09 1 1.40
3.99%
= 0.030/0
.
Similar calculations can be made for the remaining years under consideration, and such data can be tabulated as tables which show the transitions of the market percentages from one year to the next. Table I11 presents historical information on new passenger car registration TABLE I11 NEWPASSENGER CARREGISTRATION BY THE MAJOR AUTOPRODUCERS, YEARSI95 I - I 957 Auto manufacturer
Number of units
1951
1952
1953
1954
1955
1956
1957
Chrysler I I03 330 884667 I I65 357 713 347 1206195 922043 I096359 Motors FordMotors I121464 947474 1443153 1706617 I980736 1694108 1818169 GeneralMotors 2167 713 I735 694 2586 697 2006595 3639 120 3024286 2683 365 All other U.S. models 646758 561 260 514821 282520 292199 223 769 177622 Foreign models 20 828 29 299 28 961 25 385 51 658 91 042 206 827 Totals
5060093 4158 394 5738989 5535464 7 169908 5955248 5982342
Also lumped together into one classification.
299
MARKOFF CHAINS BY CAR MANUFACTURER
for years 1951 to 1957; Table IV gives the same information in percent of total new car registration; and Table V presents the percentage of gain or loss of new car registration for the major automobile manufacturers. T h e latter are based on the percentages in Table IV, taking 1951 as the starting year. TABLE IV
SAME INFORMATION AS IN TABLE 111, EXPRESSED I N yo OF TOTAL NEWCARREGISTRATIONS
% of the total
Auto ~~
manufacturer Chrysler Motors Ford Motors General Motors All other U.S. models Foreign models Totals
1951
1952
1953
1954
1955
1956
1957
21.80 22.16 42.83 12.00 0.41
21.27 22.78 41.74 13.51 0.70
20.31 25.15 45.07 8.97
0.50
12.90 30.83 50.70 4.98 0.59
16.82 27.63 50.75 3.98 0.82
15.84 28.45 50.78 3.64 1.65
18.34 30.39 44.85 2.96 3.46
100.00
100.00
100.00
100.00
100.00
100.00
100.00
TABLE V PERCENT OF GAINOR LOSS OF THE NEW CAR %GISTRATIONS FOR MAJOR AUTOMOBILE MANUFACTURERS (BASED ON PERCENTAGES IN TABLE I1 WITH 1951 AS THE STARTING YEAR) Auto manufacturer Chrysler Motors Ford Motors General Motors All other U S . models Foreign models Total increase
Percent gain or loss with respect to preceding year 1951
1952
1953
1954
1955
1956
1957
0
0
-0.53 0.62 -1.09 0.71 0.29
-0.96 2.37 3.33 4.54 -4.20
-7.41 5.68 5.63 -3.99 0.09
3.92 -3.20 0.05 -1.00 0.23
-1.34 0.82 0.03 4.34 0.83
2.86 1.94 -5.93 -0.68 1.81
0
1.62
5.70
11.40
4.20
1.68
6.61
0 0
0
Tables VI through XI offer information on the market transition for the major automobile manufacturers, expressed in percentage of the total market, on a year-to-year basis. I n Table VI the market transition is from 1951 to 1952, in Table VII from 1952 to 1953, and so on to Table XI whose data correspond to the market transition from 1956 to 1957.
300
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
TABLE VI MARKET TRANSITIONS (EXPRESSED I N PERCENTAGE OF TOTAL MARKET) FROM 195 I
C F G A 0 1951 Totals
TO
1952"
C
F
G
A
0
1952 Totals
21.27 0.20 0.00 0.24 0.09
0.00 22. I6 0.00 0.00 00:00
0.00 0.42 41.75 00.47 0.20
0.00 0.00 0.00 12.00 0.00
0.00 0.00 0.00 0.00 0.41
21.27 22.78 41.74 13.51 0.41
21.80
22. I6
42.83
12.80
0.41
100.OO~o
Key to symbols in this and following tables: C, Chrysler Motors; F, Ford Motors; G, General Motors; A, all other American makes; 0, Overseas (foreign) makes. TABLE VII MARKET TRANSITIONS FROM 1952
C F
G A
0
1952 Totals
C
F
20.31 0.40 0.56 0.00 0.00
0.00 22.78 0.00 0.00 0.00
21.27
22.78
TO
1953
A
0
1953 Totals
0.00 0.00 41.74 0.00 0.00
0.00 1.89 2.65 8.97 0.00
0.00 0.08 0.12 0.00 0.50
20.31 25.15 45.07 8.97 0.50
41.74
13.51
0.70
100.00yo
0
1954 Totals 12.90 30.83 50.70 4.98 0.59
G
TABLE VIII MARKET TRANSITIONS FROM 1953
C F G
A
0
1953 Totals
TO
1954
C
F
G
A
12.90 3.69 3.66 0.00 0.06
0.00 25.15 0.00 0.00 0.00
0.00 0.00 45.07 0.00 0.00
0.00 1.99 1.97 4.98 0.03
0 0 0 0 0.50
20.31
25.15
45.07
8.97
0.50
100.OOo/o
30 1
MARKOFF CHAINS BY CAR MANUFACTURER
TABLE IX MARKET TRANSITIONS FROM 1954
C F
G A 0
1954 Totals
TO
1955
C
F
G
A
0
1955 Totals
12.90 0.00 0.00 0.00 0.00
2.99 27.63 0.04 0.00 0.17
0.00 0.00 50.70 0.00 0.00
0.93 0.00 0.01 3.98 0.06
0.00 0.00 0.00 0.00 0.59
16.82 27.63 50.75 3..98 Oi82
1290
30.83
50.70
4.98
0.59
100.OOo/o
TABLE X MARKET TRANSITIONS FROM 1955
C F G
A 0 1955 Totals
TO
1956
C
F
G
A
0
1956 Totals
15.48 0.66 0.02 0.00 0.66
0.00 27.63 0.00 0.00 0.00
0.00
0.00 50.75 0.00 0.00
0.00 0.16 0.01 3.64 0.17
0.00 0.00 0.00 0.00 0.82
15.48 28.45 50.78 3.64 1.65
16.82
27.63
50.75
3.98
0.82
lOO.OOO/o
A
0
1957 Totals
TABLE XI MARKET TRANSITIONS FROM 1956
C F G A 0
1956 Totals
TO
1957
C
F
G
15.48 0.00 0.00 0.00 0.00
0.00 28.45 0.00 0.00 0.00
2.57 1.74 44.85 0.00 1.62
0.29 0.20 0.00 2.96 0.19
0.00 0.00 0.00 0.00 1.65
18.33 30.39 44.85 2.96 3.47
15.48
28.45
50.78
3.64
1.65
lOO.OOO/o
302
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
On the hypothesis that the available sample of six years was representative enough of the corresponding population of transition,the experimenters tested various approaches to the continuation of the work. One of the approaches involved the use of the mode or the median as the “representative” value for a transition cell. Having obtained unsatisfactory results they tested certain other “values” derived from the available data. They finally decided to experiment with the sum-total of the transition values for the same cell. By summing up the corresponding values in Tables VI through XI, the data in Table XI1 were obtained. TABLE XI1 SUMMATION OF TABLES VI
TO
XI
C
F
G
A
A 0
98.34 4.95 4.24 0.23 0.81
2.98 153.80 0.04 0.00 0.18
2.56 2.16 284.89 0.48 1.83
1.22 4.23 2.68 34.33 0.45
0.00 0.08 0.12 0.00 4.47
Totals
108.57
157.00
291.92
42.91
4.67
C
F G
0
If we let Table XI1 represent the transition from the “present year” T to the next year T 1, the summation of each of the columns gives a number representing the portion of the market that one automobile manufacturer had in relation to his four competitors. Taking year T as the reference for future years, T 1, T 2, T n, the columns of the Table in question can be expressed in a percent form. T h e transformated matrix is given in Table XIII.
+
+
+
+
TABLE XI11
TRANSFORMATION MATRIX FROM YEART C
F
G
TO
YEART
+I
A
0 .
C F G A 0
0.905 0.046 0.039 0.002 0.008
0.019 0.980 0.000 O.OO0 0.001
0.009 0.007 0.976 0.002 0.006
0.028 0.099 0.062 0.801 0.010
0.000 0.017 0.026 0.000 0.957
Totals
1 .Ooo
1 .om
1 .000
1.Ooo
1 .ooo
-
MARKOFF CHAINS BY CAR MANUFACTURER
303
T h e data in Table XI11 can also be represented in a graphical form, as in Fig. 1. Arrows indicate the direction of the change, and the corresponding percentages are duly noted.
Fic. 1
Assuming that the information presented in Table XI11 corresponds to a state of equilibrium,* from these same data we can derive a set of transition equations. For instance, let C’, F’,G‘, A’, and 0’ represent the components of a vector resulting from the previous state (C, F , G, A , 0). T h e equations derived from the information in the transition matrix, expressing the relation between the components of the two vector states, would be as follows: C’ = 0.905C F’ = 0.046C G’ = 0.039C A’ = 0.002C 0‘ = 0.008C
+ 0.019F + 0.009G + 0.028A, + 0.980F + 0.007G + 0.099A + 0.0170, + 0.9766 + 0.062A + 0.0260, + 0.002G + 0.801A, + 0.001F + 0.006G + 0.010A + 0.9570;
or the resultant vector X, is derived by multiplying the transition matrix [MI by the vector X , which is the state C, F,G, A , 0 of the system. A supposition,
16.
304
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
If X , represents the vector after n transformations, this could be expressed as X , = [MIX,-, . Based on the same suppostion, the experimenters derived simulated data regarding the percent share of the market for the competing automobile manufacturers. They then compared this information against the available historical data. T h e results of the comparison are presented in Figs. 2 through 6. Three simulated data curves are plotted in each
22 1951
1952
1953
I954
1955
I956
1957
Year FIG.2. New Ford Motors cars registered in C7.S. during years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for I-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
figure. Each corresponds to a different cycling procedure. Namely, the hypothesis was made that if other things are kept equal and the cycle of transition is varied the resulting simulated curve may approximate more closely the historical data, which in every case are represented by curve 1. Three alternative cycles have been considered, corresponding to
305
MARKOFF CHAINS BY CAR MANUFACTURER
curves 2, 3, and 4. Curve 2 is based on simulated data for a one-year cycle. For this curve the matrix of transition probabilities given in Table XIV was used. Curve 3 corresponds to simulated data for a two-year cycle, while curve 4 reflects the outcome of a simulation involving a three-year cycle. This reference to curves I , 2, 3 , and 4 is
&
1
20
In E
0 ..a-
e
c
.-In 0 2
18
\
\
L
0
u B
0)
0
16
c 0 c
+ 0 c
-
.-0
bQ
14
In Q In L
r'
2,
\
12
0
\
\ ' \
10
1951
I
1952
I
1953
I
I
I
1454
1955
1956
Year
1 1957
FIG. 3. New Chrysler Motors cars registered in U.S. during years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
valid for all figures under consideration. T h e rationale for the twoyear and three-year cycle hypotheses was that the American market is getting into the habit of changing the low-priced cars on a two-to-three years basis. T h e resulting graphical plot of the simulated data indicates that the year-to-year basis is the closest one to the actual. In Fig. 2 (Ford Motor
306
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Company), curve 2 indicates with considerable accuracy the trend of the company with respect to percent share of the market. Curve 3 gives, generally, an overestimate and only once matches the actual data, while Ford cars sales were pick. Finally, curve 4 grossly overestimates market trends.
-$ Y
50-
v)
c
0 .-c
i
‘548 ?!
’
e
L
3
Q
-
-
46
0
c
0
c
w-
0
c 0 .-
z
4-
44
Q
e0
c
0
42
c
Q,
c3 40
I
1951
1
I
I
I
I
1952
1953
1954
1955
1956
1 1957
Year FIG.4. New General Motors cars registered in U.S. during the years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles.
I n the case of Chrysler Motors, curve 3 presents the best approximation of the trend in the company’s percent share of the market. T h e approach represented by curve 2 seems to be “optimistic,” while that of curve 4 deviates considerably from the actual data. Comparing the results for Ford and Chrysler of approach 4, we see that in both cases this three-year cycling has a tendency to go to extremes; when the
307
MARKOFF CHAINS BY CAR MANUFACTURER
percent share of the market is ascending it overestimates the gains, when it is descending it overestimates the losses. It should be noticed, however, that, as considered here, the approach of curve 4 is based on a smaller sample than either that of curve 2 or 3. I n the case of the General Motors, all three approaches failed to
I--- , 1952
1953
1954
1955
1956
1957
Year FIG. 5. All other new American-made cars ,registered in the U.S. during the years 1951-1957 (all those not produced by Ford, Chrysler, or General Motors). Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for 2-year cycles, and curve 4 on simulated data for 3-year cycles. reflert t h e l a r o e inrreace in
the
r n m n a n v ’ c charp nf t h e market fnr
the
years 1954, 1955, and 1956. Curve 2 still indicates the trend in percent share over the whole six-year span without the above-mentioned gains, while curves 3 and 4 present an off-phase increase in share, and they both are sensitive in the corresponding decrease during 1957. Figure 5 presents actual and simulated data for the other American
308
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
makes. While all three simulated curves reflect the trend in the market, curve 2 is the closest to the real data. With the overseas makes, as seen in Fig. 6, curve 2 gives the closest approximation. On the basis of the foregoing, it can reasonably be concluded that Markoff chains may eventually present good planning possibilities, but more research is necessary, both to establish its areas of best potential and to develop the sensitivity of the method.
1951
1952
1953
1954
1955
l95b
1957
Year FIG. 6. New overseas (foreign) made cars registered in the U.S. during the years 1951-1957. Curve 1 is based on actual data, curve 2 on simulated data for 1-year cycles, curve 3 on simulated data for2-year cycles, and curve 4 on simulated data for 3-year cycles.
A Used-Car Model For another application of Markoff chains, an evaluation was made of the changes of automobile makes that take place when used cars are traded in for new cars. T h e obtained information is mostly from estimates made by car dealers in the Washington, D. C. area.
309
A USED-CAR MODEL
TABLE XIV CARTRADE-IN PERCENTAGES Car traded in
New car buy Chevrolet Ford Plymouth Others
Chevrolet
Ford
Plymouth
Others
0.85 0.06 0.04 0.05
0.09 0.82 0.05 0.04
0.03 0.03 0.90 0.04
0.10 0.10 0.10 0.70
T h e information in question is presented in Table XIV in the 'form
of a matrix of transition probabilities. If we assume that 100 cars of
each make: Chevrolet, Ford, Plymouth, and Others (lumped together) are traded in each year, the distribution of cars at the end of each year would be as shown in Table XV. T h e results indicate that if the trade-in percentages do not change, and if the impact of first-time buyers is ignored, Chevrolet will have greatly improved its position, and the Other makes will have very reduced sales. These calculations are made on the basis of suppositions advanced by car dealers and reflect their way of thinking and of projecting into the future of their trade. The model was developed as a minor research project in 1957, and, since then, market evolution has generally tended to support the obtained results. TABLE XV DISTRIBUTION OF CARSAFTER 1 Years
Chevrolet
TO
Ford
15 YEARS Plymouth
Others
100
100 83 73 67 62 59 56 54 53 52 52 53 54 55 55 55
~~
0
100
1
107
2 3 4 5 6 7 8 9
ill
10 11
12 13 14 15
1 I3 115 1 I7
118 1 I9 119 118 1 I7
116 115 1 I4 I I4 1 I4
100 101 101 100 100
99 98 97 95 95 94 93 92 91 91 91
109 1 I5 120 123 125 I28 130 133 135 137 138 139 I40 I40 140
3 10
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
Studying the Impact of the ABC Car on the American Economy Over a number of years, considerable discussion has taken place in the United States on the rationale of the yearly change in model by all major American car manufacturers. T h e manufacturers have strongly maintained that the yearly change in model is the lifeblood of the growth in the automobile industry, and, therefore, both in the long and in the short run it is to the benefit of the nation. T o document their reports, car manufacturers have insisted that without the yearly change in design the automobile market would limit itself to the replacement of scrapped cars, or below the fourmillion-unit mark. In contrast, automobile sales for the last eight years varied in the range of five to seven and a half million units due to the artificial obsolescence or, in different terms, to the yearly change in model. On the other side of the fence, some economists argue that these yearly model changes with the corresponding retooling etc. are a kind of waste and that the American economy can do as well without them. T o test the validity of these arguments, we will assume that a “new” automobile company would design, produce, and market the American Basic Car, hereafter referred to as ABC, with a guaranteed no-change in model for a period of twelve years.5 T h e engineering dependability of the car is also set at the same twelve-year level. By this token, a buyer of the ABC would have no inducement, either because of an artificial or real obsolescence, to trade in this car before the 12-year period is over. T h e objective is to study the effect of this policy on the American market. Say that the XBC car is built in the size range of the Chevrolet-FordPlymouth lines with an economy engine. Say also, that because of its novelty the first year of being introduced into the market it captures somewhat over 54 yo of the total car sales, dropping thereafter to 50 yo. By the same token, a certain car market of (say) 5 500 000 cars would go up to 6 OOO 000 due to ABC’s advent. Finally, it is assumed that the “other” cars are traded in every three years, as they used to up to now and that, in every trade-in model, ABC takes the 50 yo of the buyers. Some of the quantitative aspects of these assumptions may be regarded as unrealistic, yet the impact the compact cars had in the year of their introduction could help document that this is not so bad an estimate.
‘
To ease the questions which will automatically arise in regard to a marketing and servicing network, it is acceptable to assume that ABC represents the “Basic Cars” of all established manufacturers lumped together. The same twelve-year policy is assumed to prevail.
STUDYING THE IMPACT OF ABC CAR ON AMERICAN ECONOMY
311
A second tentative statement must also be made. I t concerns a static car market. With it, the result will be a major upheaval in the American economy. T h e upheaval would be reflected in the substantial shrinkage of the labor demand, a decline in the metals and other materials industries, a substantial reduction in numbers of dealers, advertisers, and auto service companies, and a decline in auto-parts manufacturing concerns. These changes would have an adverse affect on the purchasing power of the American public with an accompanying drastic shrinkage in all phases of the economy. We will assume that the “present” market is divided as follows: Northern Company Western Company Eastern Company Independents
32% 29% 25% 14%
T h e owners of Big Three’s cars, in 85 yo of all cases, trade for the same car. T h e remaining 15 yo go about even to the two other makes or to the independents. Owners of independent products stick, about 91 yo,with their makes; the remaining 9 yo distribute themselves about evenly among the big three makes. From these data, the following matrix of annual transformation can be developed:
Northern Western Eastern Indep.
Northern (N) 0.85N O.05N 0.05N O.OSN
Western (W) 0.05W 0.85W 0.05W 0.05W
Eastern (E) O.05E 0.05E 0.85E 0.05E
Indep. (D) 0.03D 0.03D 0.03D 0.91D
I n the year “zero” based on the given percentages of the market, Northern manufactures Western manufactures Eastern manufactures Independents manufacturers Total
I 760 000 Cars I 595 000 Cars I 375 000 Cars 770 000 Cars 5 500 000 Cars
In the ensuing years the market changes as follows: Year I
Northern Western Eastern Indep.
1 667 600 1 535 000
Year 2 1 588 100
I 399 000 937 200
I 484 700 1 346 100 1081 100
5 500 000
5 500 000
3 12
16.
MARKET EVALUATION THROUGH TRANSITION PROBABILITIES
T h e big three are losing ground because of the hypothesis made that they command a lower customer loyalty than the independents. Now say that into this market, at year “one,” the new car is introduced by the ABC Company. This car guarantees the stated twelve-year life and no model change during the same period. We made reference to a market of 6 000 000 units the first year and some 54 yo of the share would be gained by the ABC car the first year. For the subsequent years the matrix of transformation is Northern Western Eastern Indep. ABC
Northern 0.425N 0.025N 0.025N 0.025N 0.500N
Western 0.005W 0.425W 0.025W 0.025W 0.500W
Eastern 0.025E 0.025E 0.425E 0.025E 0.500E
Indep. 0.015D 0.015D 0.015D 0.455D 0.500D
After the first year, the hypothetical American market will continue for two more years as shown in the matrix for the pre-ABC period. TABLE XVI DISTRIBUTION OF CARSAT
THE
END OF EACH.YEAR
~~
Year
Company Northern Western Eastern Indep. ABC Total
833 800 767 800 679 800 468 600 3 250 000
1
2 794 050 742 350 673 060 540 550 2 750 000
3 761 500 720 000 665 500 620 000 2 750 000
4 368 500 350 300 327 800 328 400 1 375 000
6 000 000
5 500 000
5 500 000
2 750 000
Northern Western Eastern Indep. ABC Total
5 357 000 342 400 324 400 351 200 1375 000 2 750 000
6 347 300 335 600 321 300 370 800 1375 000 2 750 000
7 169 600 164 900 159 200 193 800 687 500 1 375 000
8 166 200 I62 400 157 900 201 000 687 500 1 375 000
9 163 300 160 300 156 700 207 200 687 500
10 80 400 79 200 77 800 106 300 343 750
11 79 350 78 450 77 200 108 750 343 750
12 78 700 77 700 76 700 110 750 343 750
1375 000
687 500
687 500
687 500
Northern Western Eastern Indep. ABC Total
STUDYING THE IMPACT OF ABC CAR ON AMERICAN ECONOMY
313
I n the fourth year, however, the ABC owners will not trade their cars as they consider them good, engineering and designwise, for 12 years. This behavior will repeat itself to the twelfth year. T h e effect of the super-durable car can be seen in Table XVI. T h e “old line” companies, which before the advent of the ABC had manufactured 5 500 000 cars per year, twelve years later are down to 343 750 cars. T h e ABC Company, while always maintaining at least 50% of the market, drops from 3 250 000 cars in year “one” to 343 750 cars in the 12th year. T h e result, therefore, of this super-durable inovation would be a virtual collapse of the industry. Though this is just a hypothesis and the chosen percentages, such as 50%, might be considered as an exaggeration, it is nevertheless true that the same result would have been obtained with small, “more reasonable” percentages] the magnitude of this outcome becoming essentially a matter of time.