7f
WORKING PAPER
ALFRED P. SLOAN SCHOOL OF MANAGEMENT
DYNAMICS OF PRICE ELASTICITY
AND THE PRODUCT LIFE CYCLE  AN EMPIRICAL STUDY*
Hermann Simon
**
WP 103578
November 1978
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
DYNAMICS OF PRICE ELASTICITY
AND THE PRODUCT LIFE CYCLE  AN EMPIRICAL STUDY*
Hermann Simon
**
WP 103578 November 1978
The author gratefully acknowledges the helpful comments of
Horst Albach, Helmut Bruse (University of Bonn), Alain Bultez
(EIASM Brussels), and Alvin J. Silk (M.I.T.).
** Assistant Professor of Management Science
University of Bonn, and
Visiting Fellow, Sloan School of Management,
Massachusetts Institute of Technology
ABSTRACT
The author presents a product life cycle model which incorporates
carryovereffects and obsolescence and allows for timevarying price
responses. An empirical study of 35 products reveals typical changes
in price elasticity over the product life cycle and casts doubt upon
the hypotheses prevailing in the marketing literature. Some important
implications for strategic pricing and antitrust issues are being
discussed.
13 LOIS
 1
INTRODUCTION
In the marketing literature it has frequently been alleged that marketing
strategy should vary over the product life cycle (Kotler 1971, Lambin 1970,
Levitt 1965, Wasson 1974, sec also Dhalla and Yuspeh 1976). Such allegations
presuppose a certain knowledge on the efficiency of various marketing instruments
at different stages of the life cycle. In fact, very little is known about this
issue. In support of the allegations, reference is usually made to Mickwitz
(Kotler 1971, p. 62; Lambin 1970, p. 15; Parsons 1975) who  back in 1959  pre
sented some theoretical considerations on the changes in marketing elasticities
over the life cycle, but did not give any empirical evidence of his hypotheses.
Too often no clear distinction between the life cycle of a particular product
and the life cycle of a whole product class has been made, two exceptions being
the studies of Polli and Cook (1969) and Dhalla and Yuspeh (1976). The present
study is clearly confined to single products, no conclusions on whole product
classes will be drawn. Throughout the paper, the term product life cycle (PLC)
denotes the time series q. ,,..., q. T of quantities sol d of a particular pro
duct or brand i. The PLCconcept is not understood as an ideal type model.
We focus on price and on the changes in price elasticity over time. According
to Mickwitz (1959) and his followers price elasticity increases over the first
three stages of the PLC (introduction, growth, maturity) and decreases during
the stage of decline.
The first part of this hypothesis seems to be supported by some findings of
diffusion research according to which early adopters of new products typically
have higher incomes and pay less attention to price than later adopters do
(Robertson 1967, Rogers 1968). The hypothesis is also confirmed by a General
Motors study on the price elasticity of automobile demand for the years 1919
 38 to which Dean (1950, p. 227) refers. One should note, however, that both
the diffusion studies and the GM study are concerned with product classes and
2 ~
do not read,,, allow conclusions for single products or brands.
AS for single products , a great many empi>ica, tests of dynamic sales respons,
functus have been conducted, almost al, of which are, however, related to
"*..** Clarke (,„, reviewed about 70 of these stuoies, further review,
can be found in Parsons and Schultz (1976) and Dh.ll. (1978).
Relatively few studies include nrirc **=„,, ,
'ciuoe price as an explanatory variable (Telser 1962
Umbin ,970, Houston .„d Weiss ,974, Wildt ,974, Lambin.Naert.and Bultez ,97,
riarty ,975, Lambin ,97S, Pr.sad ,d Ping ,g 76 , A „ of these studjes ^
—invariant price response or price elasticity coefficients .nd, therefore
do not per.it .ny conclusive inference on the cb.nges in price response or
Pnce elasticity over the PLC. The o„,y models which Include timevarying
sales responses are ,i mi ted to advertising issues (8eckwith ,9,:, P ar sons ,975
Wildt ,976, Winer ,976, Erickson ,977).
««* (,977a,b, has presented a mo del in which price elasticity varies with i
aspect to the advertising expenditure, but nevertheless is const.nt with
respect to tin*. A dec re.se in the magnitude of price elasticity over « is
Produced in the wellknown competitive simulation mode, of Kotler (,965, This
de,, however, can bard,y be tested e m pirica„y and yie,ds  due to the fact
that price elasticity approaches the zero level  stratemV » „ •
levei strategic recommendations
which cannot be considered as reasmvhia. «.■ •
as reasonable; this is shown in Simon (1978).
This short survey, thus, leads to the conclusion Mat „„ • •
uie conclusion that no convincing empirical
or theoretical evidence nf th a ~i»..
evidence of the changes in price elasticity over the PLC and
of the marketing efficiency of price at different stages of the PLC is available
THE DATA
Data on prices and quantities sold of 43 products (brands) on 7 different
markets were available for this study. All data are of most recent origin
(all after 1970) and refer to the West German market. They were supplied by
large German corporations on a confidential basis so that the product identi
ties cannot be revealed. The most important data characteristics are given
in table 1.
INSERT TABLE 1 HERE
All products represent frequently purchased branded items. On each market,
products at different stages of their individual PLC's are represented. All
markets had been established before the period under investigation so that our
analysis applies and is limited to products which are introduced onto markets
with existing substitutes, it does not apply to generically new products. We
are not aware of any single study which includes a greater number of products.
The data show enough variation to admit an examination of the dynamic relation
ships between prices and sales. The managers concerned with the products con
sider price (besides quality which remained unchanged over the period under
investigation) as the most important marketing variable.
Even in the case of the detergents, the absence of nonprice data
doesn't seem to be too serious a problem. This is in particular true for
advertising data due to two reasons. On the one hand, advertising is
much less important in Germany than in the U.S.; this is mainly due to
strict limitations of TVadvertising (only 20 min. per weekday, no adv. on
Sundays and holidays; in 1977 the advertising budget of Procter & Gamble (USA)
alone amounted to 93.8% of the total amount spent on TVadv. in Germany).
 4
On the other hand, the managers hold that advertising spending for detergents
is rather evenly distributed over the year and hasn't changed much over the
period under investigation, so that the impact of advertising is likely to be
adequately reflected in the constant term of the sales function.
MODEL SPECIFICATION
The empirically tested dynamic sales response models usually have the form
1i,t = a l +a 2 Vt1 +f(p i,t' Pi,t } (1 >
where q. . product i's sales in period t (either units or market share)
p.; t product i's price in period t
p. t some weighted average price of products competing with
1,1.
product i in period t
f(») price response function
a. , ao parameters
The sales and price variables are either in natural or in logarithmic
dimension. Typically all functional relationships in (1) are assumed to be
timeinvariant. Hence, for constant prices and a 2 ! < 1, function (1) can only
describe the approach of q. . towards an equilibrium level of sales. The
1 1*
dynamics of (1) do not allow for a representation of a life cycle curve with
an ascending and a descending branch if prices remain unchanged. Moreover, the
timeinvariant price response presupposed in this function must be considered
as a very restrictive assumption.
Within the last few years a number of advertising models which allow for time
varying coefficients f both advertising and the lagged sales variable, the
so called "carryover effect", have been proposed (Beckwith 1972, Parsons 1975,
Wildt 1976, Winer 1976). The results of these few studies as to the carryover
effect are not unequivocal Parsons (1975), for instance, presupposed
an increase in the carryover effect over time and Wildt (1976) investigated
industry sales and not product sales. The results of Beckwith (1972) and
Winer (1976) both ofwhom studied the Lydia Pinkham data indicate a downward
tendency of the carryover effect. Product life cycle theory indeed suggests
that the abilitiy of a product to retain its customers from period to period
should decrease in the course of time due to the introduction of new competit
ive products which, in a dynamic market, are likely to be superior either tech
nologically or "psychologically" (fashion, taste etc.). The erosion or
"obsolescence" of the old products and the diffusion of the new products, how
ever, occur gradually and not immediately. It seems reasonable to assume an
exponential pattern of the decrease in the carryover in order to account for
this phenomenon.
Thus, we obtain for the nonprice terms in (1), for which we write A. .
i , t
Al: A 1§t  a 1 + a 2 .q. jt _ 1 . (l^^i ( 2)
where < a 3 < 1 can be interpreted as'rate of obsolescence' and t. denotes the
period of introduction of product i. For t=t. we have A. .=8,, hence a, repre
sents product i's initial demand potential.
The results of Winer (1976) indicate that not only the carryover effect but
also the initial demand potential may be subject to the obsolescence phenome
non. Assuming the same rate a 3 we obtain as an alternative model to (2)
A2: A. )t = (a^a^.^) (la/^l (3)
It should be noted that Al and A2 include the function with constant parameters
as a special case where ao=0.
A great variety of possible life cycle curves can be represented by means of
these simple functions. This flexibility is highly important since empirical
PLC's tend to have very different shapes (Cox 1967, Polli and Cook 1969, Wasson
1974, Dhalla and Yuspeh 1976). Figure 1 gives an illustration of this flexi
bility [f (.)<>].
INSERT FIGURE 1 HERE
Some of the products under investigation show seasonal sales patterns which are
due to seasonrelated diseases in the case of the drugs and to certain habits
of German housewives in the case of the detergents (draperies etc. are typically
laundered in spring and fall). Both managerial experience and visual inspection
of the sales curves indicated that only two types of seasonal patterns existed
so that one dummy variable D. = {0,1} is sufficient to account for the
seasonalities. Adding the seasonal term to Al and A2 respectively we obtain.
A3: A. jt = a x + d.D t + a 2 q^^lag)*"*! (4)
A4: A i)t = ( 3l + dD t + a 2 q 1>t>1 ) (la 3 ) t " t i (5)
In a few cases, a further version A5 which is equal to Al with a, =0 has been
tested.
It seems reasonable to assume that product i's sales depend both on the absolute
level of its price p. . and on the differential between p. t and the prices of
competing products.
In the absence of evidence to the contrary, we hypothesize and test a linear
relationship between q. . and the absolute price p. ..
B i,t = b Pi,t < 6)
As to the sales effect of the price differential we adopt a hypothesis which
was first proposed by Gutenberg (1955, 1976) and has found wide acceptance in
the European marketing literature. According to this hypothesis a relatively
small price differential is assumed to have an underproportional sales effect,
whereas a relatively great price differential is assumed to produce an over
7 
proportional sales response. This hypothesis is based on the experience that
only very few customers are likely to switch from their accustomed brand to
another brand if the price differential changes by e.g. 1% or 2% only, whereas
the number of brand switchers typically grows overproportionally when the
price differential increases for instance to 20% or 30%.
A nonlinear relationship of this type can be represented by a sinhfunction
(sinus hyperbolicus, Albach 1973). We consider two versions of sales response
to price differentials, the first being
CI: C. )t = cjsinh ( ^ Ap i>t ) (7)
where Ap. = (p n  t p. + )/P, t is the price differential,
Y m. .p. . is the market share (m. .)
i»t " ._, ,, n weighted average price of
3?i J ' products competing with i,
c, ,C2 are parameters.
In the version CI the price response is timeinvariant. The second version
to be tested is based on the assumption that the sales response on a price
differential is proportional to the total market demand hitherto effective.
C2: C. t = c x sinh (c 2 AP i>t ) q^ (8)
where n
q. , = I q. . , is the total market demand in t1 .
The version C2 meets in particular the requirement of Parsons and Schultz
(1976, p. 158) that a timevarying response should rather be explained by
marketing variables than merely by time.
The terms A, . , B. . , and C. t can be linked either additively or multiplicative
ly. We hold that a multiplicative linkage is less appropriate in our case since
it implies that the price response, i.e. the derivative 8q. + /9p 1  t , develops
proportionally with the nonprice term A. . so that the price response would
be affected by the obsolescence effect in the same way as the carryover
effect. This would, in fact, amount to a predetermination of the question
to be investigated. Therefore, the assumption of independence between the
nonprice influences and the price influences is made so that a linear
function is obtained.
<i,t =A i,t +B i,t +C i,t +u i,t < 9 >
where A. . is either Al , ,A5; C. . is either CI or C2; and u. t is the
error term.
In anticipation of the detailed regression results we note here that
the influence of the absolute price, bp. . , did not prove significant for
any of the products. This result coincides very well with the managerial
opinion that primary demand for the products under investigation has not been
affected by changes in the absolute price levels (since 1970). This applies
both to the detergents and to the pharmaceuticals.
Due to this outcome, we can confine subsequent attention to A. . and
C. .. The solid line in figure 2 gives a graphical illustration of the price
response function (with A, .=1, B. .=0, c,=.l, c 2 =10, p. .=1)
INSERT FIGURE 2 HERE
The price elasticity denotes the percentage change in sales induced by an
incremental (or 1%) change in price and is mathematically defined as
e i,t = 9 Vt /3p i,t ' p i,t /q i,t < 10)
For the two versions CI and C2 of our price response function we obtain
9 
CI
P i,t
e i t = ~ c l c 2 cosh ( c 2 Ap i t^ '~ ( 1] )
C2:
q i,t p i,t
e i>t =  Cl c 2 cosh(c 2 Ap i>t )^^l (12)
'i,t K i,t
The equations (10)  (12) show that the dimensions of prices and quantitities
are eliminated when e^ t is computed. Hence, price elasticity is a dimension
less measure of price response and can readily be compared for different products.
The proposed price response function and its price elasticity have the follow
ing properties:
(1) The function gives economically reasonable values within a certain inter
val only. It doesn't make any sense to compute the expected sales effect of
an arbitrarily large price differential (e.g. 1000SQ by means of this function.
According to Kotler (1971) this property applies to most marketing response
functions.
(2) The magnitude of price elasticity increases for increasing positive and
negative deviations of p. . from p. . ; this is a necessary consequence of
our basic assumption that sales response increases overproportionally with Ap. t
I * w •
The price elasticity values are given by the dotted line in figure 2.
(3) The function allows for any development of price elasticity over time;
£; t may decrease, increase , remain constant, or develop irregularly over
time. Some examples which give evidence of this flexibility are depicted in
figure 3 (the parameter values can be found in table 2).
INSERT FIGURE 3 HERE
(4) Since the absolute price level has turned out to have no significant
influence on sales, the direct price elasticity e. . , the crossprice
l tt
C  
elasticity e. . = 3q. ./ap. . • p. ./q. ., and the respective market share
10
elasticities have the same magnitude. Therefore, we need not distinguish
between direct and cross elasticities (though they have different signs)
and can confine ourselves to the discussion of their common magnitude.
REGRESSION RESULTS
Since market shares do not necessarily show a PLCpattern (e.g. if market
sales and product sales develop proportionally^ m. .=const.) sales units were
considered as the more appropriate dependent variable for our purpose.
The different versions of (9) are nonlinear with respect to the obsolescence
parameter a 3 and the price parameter c 2< Therefore, the nonlinear least squares
estimation technique of the TSPprogram (a GaussNewton algorithm) was applied.
The results of these estimations, however, proved highly unsatisfactory due to
the following reasons (ranked according to their importance):
 though convergence was achieved in most cases the coefficients were almost
invariably insignificant.
 the rate of obsolescence a 3 often had a negative sign which is economically
unreasonable since it implies an unlimited growth of the carryover effect.
 in about 20% of the cases no convergence was achieved.
These results suggested to attempt a different approach in which a 3 and c 2
were prefixed so that the sales function became linear in the remaining para
meters and ordinary least squares (OLSQ) estimation procedures could be applied.
The search for the obsolescence parameter a 3 was limited to the interval (0, .1)
since a 3 can reasonably be assumed not to exceed .1 for the given data inter
vals (quarters and bimonths).
A similarly apparent interval for reasonable values of c 2 is not available. For
a given Ap. ^, this parameter determines the magnitude of the argument of sinh and
11
thereby, the degree of nonlinearity of price response. One can easily realize
this relationship in figure 2 by considering Ap^ t as given and c 2 as variable.
For c 2 Ap i t  <1, sinh is almost linear; for c 2 Ap i t  > 1, sinh becomes
increasingly nonlinear. Thus, by prefixing different values of c 2 we can
account for different degrees of nonlinearity in the sales response to price
differentials.
In the estimations we usually prefixed three values in the following way
(x denotes the maximal magnitude of Ap. . over all periods)
case
value
range of
maximum
competitive price effect
of c 2
argument
of sinh
of sinh
(shape of sinh within range)
1
c 2 =l/x i
1 +1
1.17
quasilinear (proportional)
2
c 2 =2/x i
2 +2
3.62
medium nonlinear
3
c 2 =3/x i
3 +3
10.01
highly nonlinear
In this way, both a quasilinear and various nonlinear patterns of sales
response to price differentials were admitted. In a few cases, where the
results indicated that smaller or greater values of c 2 would improve the
estimation some additional prefixations of c 2 were tested.
For each product, about 20  25 estimations with different combinations of a.
and c 2 were run, the total number of regressions amounting to about 5000. The
detailed results are reported in table 2.
INSERT TABLE 2 SOMEWHERE HERE
(Footnote to table 2)
Column (1) gives the product number (first digit: market, second digit: pro
duct). 0LSQ in column (2) means ordinary least squares and C0RC stands for
the CochraneOrcutt iterative technique  a generalized least squares
method  which was applied when the DurbinWatson statistic (DW) of the 0LSQ
12 
estimate fell into the inconclusive range or indicated autocorrelation. This
enforced criterion has been suggested (Schneeweiss 1974, p. 244) since DW is
of limited reliability when one of the regressors is the lagged dependent
variable (Durbin 1970). Durbin's H which would be appropriate in this case
is not provided in the TSPprogram of MITHarvard by means of which the
estimations were made.
Column (10) gives the introduction periods l. t a negative number indicates
that the product has been introduced before the period under investigation.
In the cases marked by an asterisk the true introduction periods were not
available, and t. was set equal to 1. The numbers in parentheses are the
tstatistics and a, b, c, and d denote significance at 1%, 5%, 10%, and 25%
respectively (one tailed test).
(End of footnote table 2).
Reasonable results have been obtained for 35 out of the 43 products. A summary
of the statistical criteria of the regressions is given in table 3.
INSERT TABLE 3 HERE
Thus, 82% of the coefficients were significant at 90% or more and 83% of
2
the coefficients of determination R exceeded 0.60. These results give strong
empirical support to the hypotheses underlying our model. Both the PLCdynamics
and the competitive price effects appear to be adequately represented.
PRICE ELASTICITIES
From the regression equations, we computed price elasticities for all products
and all periods. For this purpose the actual values of prices and quantities
were inserted into (11) and (12) respectively.
 13
In order to obtain condensed and comparable measures of the magnitude and the
development of each product's price elasticity the median i and the average
growth rate g of each time series e. . , t = t. , ,T were calculated. In
this case, the median is the appropriate measure of the average magnitude of
price elasticity since it excludes the influence of outlyers which were not
infrequent. The average growth rate g is obtained as the geometric mean of
the time series of elasticity growth rates. Note that the arithmetic mean
would be inappropriate when applied to growth rates. The values of e and g
are given in columns (3) and (4) of table 4.
INSERT TABLE 4 SOMEWHERE HERE
One readily recognizes from column (3) in table 4 that the elasticity medians
of the two product groups are considerably different. Almost all of the price
elasticities of the pharmaceutical products (markets 1  4) are smaller than
(or close to) 1, whereas the values for the detergents without exception are
greater than 1. This important finding is further clarified in figure 4
where the distributions of the elasticity medians are depicted, separately
for the two product groups. Only cases with significant price influence are
included in figure 4.
INSERT FIGURE 4 HERE
The graphical illustration gives even stronger evidence of the differences
in price response between the two product groups, the medians of the two
distributions (.44 and 1.88) being significantly different at the 1%level.
Both these differences and the absolute magnitudes of price elasticities
coincide very well with the managerial experience. The results are also in
good accordance with the findings of other researchers (Telser 1962,
Lambin 1976).
 14 
The average growth rates g in column (4) of table 4 indicate that the price
elasticities have frequently undergone considerable changes over time of both
positive and negative sign. In order to investigate this issue more deeply
and to find out whether the changes in price elasticity show characteristic
linkages with certain PLCstages, we make two types of comparisons.
We first compare the elasticity growth rates of those products which were at
the same PLCstage (introduction, growth, maturity, or decline) during the
last quarter or bimonth under investigation.
In addition to this crosssection comparison we study the magnitudes of price
elasticity of one and the same product at different stages of this product's
PLC  Tnis longitudinal comparison is necessarily limited to products whose
sales curve includes at least two PLCstages; 30 products belong to this group.
Both the crosssection and the longitudinal comparisons require a preceding
classification of the actual sales curves into PLCstages. It is certainly
desirable to use objective criteria for this classification. Respective attempts,
in which growth rates, moving averages of 2, 3, and 4 growth rates, changes in
signs of growth rates, or the stage identification criteria proposed by Pol 1 i
and Cook (1969) were used, did, however, not prove useful. Polli and Cook state
themselves that their criteria "are by no means flawless" and their application
would, in fact, have led to stage sequences like e.g. maturitygrowthdecline
maturity. The growth patterns in our sample (and probably empirical growth
patterns in general) are somewhat different from the regular PLCschemes usually
found in marketing textbooks. Positive and negative growth rates or averages of
growth rates actually occurred at all stages, and the magnitudes of growth rates
showed enormous irregular variations (see also Dhalla and Yuspeh 1976).
Therefore, a standardized classification scheme was not considered as appropriate
and we decided to effect the necessary classification on the basis of a visual
inspection of the sales curves. The procedure is demonstrated for three of the
15
products under investigation in figure 5.
JNSERT FIGURE 5 HERE
Though this method may seem somewhat arbitrary we consider it as justified and
appropriate in this case. On the one hand, the resulting classification is not
likely to differ significantly from person to person, as discussions of the
author with both managers and scientists have shown. Even if there are slight
deviations in the classification they are not likely to affect the results. It
should also be noted that this way of classification fully corresponds to the
way in which the manager has to determine at which stage of its PLC a product
actually is.
To a certain degree, the appropriateness of our classification is confirmed by
a comparison of the relative average duration of each stage with the frequency
distribution of stages obtained by Pol 1 i and Cook (1969) for brands. This com
parison reveals a considerable conformity.
Introduction
Growth
Maturity
Decline
Relative average duration (%) 11.2
29.2
33.1
26.3
Frequency distribution (%)
Polli and Cook (1969) n ' a '
37
36
27
The results of the crosssectional and the longitudinal comparisons are
summarized in table 5 and columns (5)  (12) of table 4 respectively. Table
5 gives the average growth rates of price elasticity of all products arranged
according to their PLCstages during the last quarter or bimonth under investi
gation.
INSERT TABLE 5 HERE
Some striking characteristics are revealed:
 the magnitudes of g show a considerable uniformity within the various stages,
 all signs of g within the growth stage are negative,
 all signs of g within the decline stage are positive,
16 
 with only two exceptions (5.1 and 7.3) the following relation proves true
^Growth '" Maturity <s ^Decline
Thus, we can conclude from the comparison of the price elasticities of various
products being actually at different stages of their life cycles:
(1) Changes in price elasticity over the PLC seem to have a rather uniform
pattern.
(2) Price elasticity of growth products decreases over time.
(3) Price elasticity of decline products increases over time.
(4) The rates of change in price elasticity are not uniform in sign for
products being at the maturity stage. These rates, however, seem to be
smaller in magnitude than both the rates of growth products and decline
products.
In columns (5)  (12) of table 4 the numbers of quarters or bimonths and
the elasticity medians of the different PLCstages are given for each product.
If we compare for each product the medians of adjacent stages (thus, only
products with at least two stages are included), the following relationships
are revealed:
(1) In 18 out of 19 cases (95%) the relation e Introduction > e Growth is
confirmed.
(2) In 10 out of 14 cases (71%) the relation e Growth > e M a t U ritv is confl ' rmed 
(3) In 8 out of 8 cases (100%) the relation e Matun  ty < decline is conf " irmed 
The plot of the medians of the various stages further elucidates these findings.
INSERT FIGURE 6 HERE
We can summarize our findings as follows:
An empirical investigation of 35 products gives strong support to the hypo
thesis that price elasticity shows typical changes over the product life
cycle. During the introduction and the growth stage, a considerable decrease
17 
seems to prevail. At the maturity stage, price elasticity typically reaches a
minimum which is again followed by an increase during the decline stage.
These empirical findings are in contradiction to the hypotheses prevailing
in the literature (see introductory section). This contradiction may partially
be explained by the fact that usually no clear distinction between the
absolute sales effect of a price change, which is given by the derivative
3q i t/3P.j t > and the relative sales effect, which is equal to the elasticity
e i,t = 9q i,t /9p i,t' p i,t /q i,t has been made 
How can the uniformity of the empirical outcomes be explained in view of
the fact that the underlying price response function explicitly allows for
different development patterns and does not constrain the results to be as
reported. The main reason for the farreaching uniformity of the elasticity
developments has to be seen in the changes in q. . (appearing in the
denominator of the elasticity term) which typically turned o u t to be considerably
greater than the changes in the derivative and in p. . .both appearing in the numerator
1 ,u
of the elasticity term. Thus, in a certain sense the development of the sales
q. t tends to determine the changes in e. . . Though the derivative 3q. +./8p. +
typically also increases over the ascending branch of the PLC this increase
is almost never so great as to neutralize the reciprocal effect of the growth
in sales.
IMPLICATIONS
Since it has been our main objective to measure price elasticity and its changes
the managerial and antitrust implications of our findings shall be outlined in
short only. The results seem in particular important for the optimization of the
pricing strategy over the life cycle. The optimal pricing strategy is obtained
by maximizing the sum of the discounted cash flows over the periods t,...,T (the
18 
product index i is subsequently omitted)
max , = I {p t+T q t+T  C t+T (q t+T )} (l + i)" T (13)
T=U
where C(q) is the cost function and i is the discount rate.
The maximization of (13) requires a hypothesis on the presumable reaction of
competitors to the firm's price setting. This complex issue cannot be dis
cussed in great detail here. It seems, indeed, of minor importance in this
case since we are interested less in the absolute levels of optimal prices
than in their developments over time. Whereas the former are certainly
governed by the competitive reaction pattern the latter are more likely to
depend on the changes in price elasticity and crossprice elasticity over
time.
Therefore, we consider the assumption that the prices of competing products
are treated as givens and not as functions of p t as not too restrictive
for our purpose, which as aforementioned is to gain insights into the
development of optimal prices.
Under this assumption the differentiation of (13) with respect to p t leads
to the first order condition
a _ 3q. Tt 8q.
% "it + <>V c ;> W t * Mphx  C i +I ) apf 1 d + ')" T = ° (14)
where C' denotes marginal cost.
Due to the formulation of A. . in (3) and (4) we obtain the longrun effect
of a price change in t as the product of the shortrun price response, i.e.
the derivative 9q t /8p t and the cumulative carryover effect.
!!!*+!. !ft a X (l r t + T(Tl)/2 (15)
ap t 3p t a 2 [l a 3>
19
Inserting (15) into (14), multiplying by P t /q t » and solving for the optimal
price pj gives
"*t ■ TTT <=i " T^T X "WW "S (l. 3 ) rtH < rt > /2 (Itlf* (, 6 )
t t Tl
Since e. still depends on p t (16) doesn't allow for a straightforward com
putation of pi. The equation clarifies, however, the following relations:
(1) The optimal dynamic price pi is a compound of the optimal static price,
which is given by the first term in (16)  this is the wellknown Amoroso
RobinsonRelation  and the present value of the future marginal revenues
caused by a price change in t.
(2) If a 2 > 0, < a 3 < 1 , and p. + > C. + , this present value is positive
and pi is in all periods x < T less than the optimal static price (note that
this statement doesn't depend on the assumption on competitive reaction).
(3) If the price elasticity behaves according to our empirical findings
(depicted in figure 6) then the optimal markup factor £4/(1 + O is relative
ly smaller at the introduction and growth stage and relatively greater at the
maturity stage, it again decreases during the decline stage.
(4) Both the longrun price effect and the development of price elasticity
give support to a strategy of the penetration type. One should keep in mind,
however, that these statements (and our analysis as a whole) apply to pro
ducts which enter onto a market with existing substitutes and have to be
viewed under the limitations of the assumed competitive reaction pattern. The
assumption of a different pattern may considerable damp (though not eliminate)
the outlined trend in optimal prices.
New products which establish a new market or product class and, thus, have
no substitutes at the time of their introduction are in a completely different
20 
situation and, consequently, different strategic recommendations apply (see
Simon 1976).
It should also be mentioned that changes in cost have, of course, the same
importance for the pricing strategy as the price response factors. If, for
instance, marginal cost decreases according to the experience curve concept
(Henderson 1972) the optimal prices need not increase over time since the
increase in the markup factor can be compensated (or even overcompensated)
by the decrease in marginal cost.
The optimal pricing strategy for a particular product at a particular time
depends on the relative magnitudes of the demand and cost factors. Therefore,
no general recommendation as to which type of strategy is optimal can be
given, this decision has to be made in each individual case.
The numerical optimization of the pricing strategy is best achieved by means
of a branchandbound algorithm which optimizes over a finite number of price
alternatives within a prefixed price range. In figure 7 the optimal pricing
strategy for product 4.2 of our sample is depicted. The actual price of this
product remained constant at .71 whereas the price differential Ap. , being
negative for all t, changed from  48% at t=l to  26% at t=10. The firm
under consideration usually prices its products above the average prices of
competing products. The competitors presumably expect this behavior and are
unlikely to react if prices are up to this expectation.
Therefore, the optimization was run over the interval (.48, .80). The marginal
cost was assumed to be constant (CI = .20) and an annual discount rate of
10 was applied, this rate is actually used in investment decisions by the
producer of the product. The optimization was carried out for a planning
horizon of 10 quarters or 2 1/2 years.
INSERT FIGURE 7 HERE
 21
The resulting optimal strategy confirms the conclusion drawn from equation (16)
The initially prices are considerably lower than the prices in later periods
(penetration strategy). The fact that the initial prices are also less than
the actual prices may be an indication that practitioners don't pay sufficient
attention to the longrun effects of pricing. The present value of profits of
the optimal strategy exceeds the respective value of the actual strategy by
33.7%.
The limitations of such an optimization have, of course, to be observed. Our
model doesn't incorporate any negative goodwill or sales responses which may
result from the price increases, the necessity to raise prices several times
may well prevent managers from setting a low introduction price. Such con
siderations can, however, hardly be represented in a quantitative model
and should have their proper place at the stage of managerial evaluation of
the optimization results.
Further implications of our analysis refer to antitrust issues. The question
whether price competition is workable or not and whether dominant products
are subject to substantial competition or not played an important role in a
number of recent antitrust cases (both in Germany and in the European
Community).
The discussions on these points have regularly been characterized by a lack
of objective information. The methods described in this article represent an
appropriate tool for the measurement of competitive intensity and interde
pendences under dynamic conditions. Albach (1977) used similar tools to
determine the relevant market for pharmaceutical products and to measure the
effectiveness of competition. He also extended the concept of the dynamic
crossprice elasticity by estimating partial crossprice elasticities between
single products or product groups. In this way an objective assessment of
a product's competitive position on a dynamic market seems attainable.
22 
SUMMARY
A dynamic sales model which incorporates the product life cycle concept and
timevarying price responses has been presented. The model is of a very
general nature and includes both timeinvariant and timevarying carryover
effects as well as quasi linear and nonlinear patterns of sales response to
price differentials.
An empirical study of 35 products reveals typical changes in price elasticity
over the life cycle and gives support to the conclusion that the magnitude
of price elasticity decreases over the introduction and growth stage, reaches
its minimum at the maturity stage, and again increases during the decline
stage.
Though the analysis is subject to limitations (e.g. relatively short periods
under investigation, many products included only 2 or 3 PLCstages) the results
cast heavy doubts upon the hypotheses prevailing in the marketing literature.
They also call for further research for different product classes.
The findings seem to indicate the optimal ity of a penetration type strategy
for products which are introduced onto markets with existing substitutes.
Further implications are related to antitrust issues.
23 
TABLE 1: DATA CHARACTERISTICS
Market
Product
Class
Number of
Products
Share of total
market repre
sented in last
period
Maximal
number of
observa
tions
basis for
price
compari
son
period
length
1
Pharma
8
70.2 %
24
weight
quarter
9
ceutical
ii
6
83.0 %
24
daily dose
quarter
3
• ii
6
84.9 %
24
daily dose
quarter
4
h
8
82.2 %
24
daily dose
quarter
5
Detergent
5
69.7 %
18
weight
bimonth
6
H
5
55.5 %
18
package
bimonth
7
Household
Cleanser
5
65.3 %
14
weight
bimonth
Pro
duct
Es t ima
t ion
Method
Func
t i on
Type
absolute
term
a l
season
dummy
d
reten
tion
ra te
a 2
obso
les
cence
a 3
price d
t i on e
c l
?v \ a 
ffect
c 2
period of
intro
duction
t
R 2
DW
1.1
OLSQ
A3/C2
69o .
(i.;?) D
447
(2.57) a
85 a
(4.67) a
.03
■ 15 a
(3.53) a
2.28
.7604
2.07
1.2
CORC
A3/C2
32.3
(1.07)
76 a
(3.!5) a
125 a
(5.54) a
.07
.0012
(.138)
2.48
10
.7104
2.08
1.3
CORC
A3/C1
144
(3.84) a
19 H
( .87)°
1.18
(5.16) a
.05
12.77
(3.33) a
8.05
5
.4372
1.86
1.4
OLSQ
A1/C2
67 a
(3.33) a

.426
(1.83) c
.04
.0125
(l.B4) C
2.34
15
.7357
2.63
1.5
0L5Q
A3/C2
5.9
(.45)
16. 3 d
(145)°
.626
(5.19) d
.01
.0099
(2.63)°
3.53
18
.9718
2.73
1.7
OLSQ
A3/C2
181 .
(2.62)°
121 a
(3.10)°
98 a
(3.71) a
.06
.0026
(1.46) C
2.02
13
.7260
2.08
2.1
OLSQ
A1/C2
466
(3.98) a

1.06
(13.43)
.01
.001
(1.81 ) c
9.42
19
.9934
2.15
2.2
OLSQ
A3/C1
641
(1.74)"
221 .
(2.40)°
132 a
(14.72) a
.01
808.6 .
(1.72) D
.538
16
.9734
2.43
2.3
OLSQ
A3/C2
141 K
(2.44) D
201 .
(2.27) D
1.22
(U.29) a
.001
.0619
(1.57) c
.525
16
.9824
1.97
2.4
CORC
A3/C2
4567
(12.81)
681
(9.32) a
.307 _,
(2.58) a
.002
.0702 ,
(.894)°
.523
32
.7847
2.21
2.5
corc
A5/C?


.88
(1.42) c
.04
.131
(1.50) C
1.39
5
.8376
1.63
2.6
OLSQ
A3/C2
64 K
(2.!9) n
15? h
(2.17) D
.978
(15.6) a
.0015
.0007
( 65)
7.06
12
.9770
2.32
3.2
CORC
A1/C2
19420
(4.20) a

.336 .
(.812)°
.04
.002
(.448)
7.32
.9801
1.79
3.3
CORC
Al/Cl
7349
(3.44) a

.35
(1.61)
.004
1196 .
(2.05)°
2.16
24
.6675
2.01
3.4
CORC
A6/C2

1.03 ,
i . * . . O
{'■LCI
.005
.0007
(.MS)
4.85
11
.9477
2.66
3.6
CORC
A5/C2


.81
(15.2) a
.003
.0013
(6.45) a
14.74
14
.9131
1.24
4.1
OLSQ
Al/Cl
6834
(4.46) a

.468
(6.24) a
.04
378
(2.65) b
3.35
17
.9693
3.09
4.2
OLSQ
Al/Cl
936 ,
(2.42) b
.756
(3.14) a
.04
45.7
(1.51) c
6.71
14
.8888
1.01
4.5
OLSQ
A1/C2
57?
(.268)

111 a
(4.04) a
.01
.0224
(1.66) C
5.66
1
.5074
2.14
4.6
OLSQ
A1/C2
9494
(7.84) a
39 a
(4.02) a
.01
.083
(5.01) a
3.85
3
.8594
1.20
4.7
OLSQ
A1/C2
1602 .
(1.79) c
.564
(2.91) a
.01
.0005
(.253)
15.58
14
.5215
2.40
4.8
OLSQ
A1/C2
2304 a
(17.1)
.801 _,
(11.26) a
.04
.007 7
(1.40)'
3.45
.9348
1.91
5.1
OLSQ
A3/C2
1160
(19.08) a
73 h
(1.80) D
.38
(2.81) 3
.08
.0104
(4.11) a
43.09
1*
.6272
1.56
5.2
OLSQ
Al/Cl
1142 ,
(9.91) a

.136 ,,
( 99) d
.08
100.2 r
(1.40) c
9.02
1«
.1102
1.85
5.3
OLSQ
A3/C2
1097 .
(2.15) b
175
(2.52) a
.186
( .".7)
.01
.0006
44.56
1*
.3788
1.85
5.4
CORC
A3/C2
782
(2.92)
125
(2.86) a
.28 .
(1.12) d
.045
.0068
(1.37) c
27.70
1*
.6356
1.87
5.5
CORC
A4/C2
196 a
(5.00) a
37 a
(3.19) a
.463^
(4.09) a
.045
.0166
(15.25) a
6.65
1
.8537
2.29
6.1
OLSQ
A3/C1
596 ,
(2.74) a
92.8 .
(2.16) b
.264
(1.06)°
.0015
126.8 .
(2.06)"
12.36
1
.5536
1.28
6.4
CORC
A3/C1
750
(3.39) a
98
(2.83) a
.108
(■45)
.005
79.7
(133)°
16.13
l m
.6074
2.04
6.5
OLSQ
A4/C2
130 b
(2.29) R
32 a
(4.62) a
.42
(175) C
.015
.0037 .
(3.H2) a
11.26
!•
.6772
2.42
7.1
OLSQ
A1/C2
1023 b
(2.06) D
.63 
(3.18) a
.01
.0028 r
(1.72) c
92.11
" 1*
.6192
1 32
7.2
OLSQ
A1/C2
445 b
(2.08) D

.75 .
(6.33) a
.01
.0169 d
(1.30)°
5.32
5
.8907
2.33
7.3
OLSQ
A1/C2
2396 ,
(3.70) a
.164 d
(.976)°
.0025
.0927 .
(4.80) a
8.85
1*
.7709
1.82
7.4
OLSQ
Al/Cl
4301 .
(2.62)°
• 79 a
(4.59) a
.01
228 .
(1.48) c
20.51
r
.6777
1.78
7.5
OLSQ
A1/C2
1784
(2.86) a
.64
(2.44) b
.03
.0494
(2.02)b
6.58
i*
.6992
1.87
25
TABLE 3: SUMMARY OF STATISTICAL CRITERIA
Coeffi
cients
Signi
1 %
ficant
5 %
Coeffic
10 %
ients
25 %
Equa
tions
<.60
R 2
.60
.70
.70
.80
.80
.90
>.90
n
118
57
23
17
9
35
6
8
6
5
10
%
100
48
20
14
8
100
17
23
17
14
29
TABLE t: PRICE E L A S T 1 CITIES
 26
(1)
Pro
duct
(?)
intro
duc t ior
period
">) /e) a) m m — («j) u\\ M)
PRICE ELASTICITIES
TTT5
price
influence
significant
at (I)
Tota
£
1 period
growth
rate g
I n tro
duc t ion
n e
Growth
n c
Mat
n
uri ty
c
Oec
n
1 i ne
c
1.1
.37
2.90
4
.33
7
.33
8
.36
5
.46
1
1.2
10
.15
1.54
2
.29
2
.22
8
.11
3
.16
ri.s.
1.3
5
.73
11.72
2
3.93
5
2.79
12
.64
1
1.4
15
.83
5.93
3
.98
7
.66
10
1.5
18
1.41
18.90
2
2.34
5
1.24
5
1.7
13
.84
15.14
3
2.35
9
.74
10
2.1
19
1.26
10.39
2
1.26
4
1.24
10
2.2
16
.34
9.41
14
.53
10
.26
5
2.3
16
.41
4.35
21
.43
3
.39
10
2.4
32
.05
10. 11
24
05
20
2.5
5
.37
12.07
5
1.68
15
.33
10
2.6
12
.25
19.71
2
2.48
11
.22
n.s .
3.2
.36
5.35
7
.73
17
.31
n . s .
3.3
24
.45
3.50
14
.34
10
.62
5
3.4
11
.14
2.14
2
.15
12
,13
n.s ,
3.6
14
.54
12.95
2
1 .62
9
.54
1
4.1
17
3.52
12.43
2
7.11
6
3.45
"
5
4.2
14
1.09
19.89
2
6.88
10
1.01
10
4.5
1
.63
1.14
3
.55
5
.67
13
.63
3
.77
10
4.6
3
.87
.66
6
.89
10
.79
6
.96
1
4.7
14
1.03
6.24
1
34.0
2
1.13
8
.94
n.s.
4.8
.43
4,71
2
.19
7
.41
14
.44
10
5.1
1«
2.85
5.50
18
2.85
1
Li_
!•
1 .22
1.10
18
1 .22
10
5.4
1»
1.07
14.28
7
4.81
11
.94
10
5.5
1"
2.22
4.75
2
1.68 
?.25
1
6.1
1«
1 .92
1.17
18
1.27 j
■
6.4
1*
1.83
3.34
7
2.18
11
1 .61
1 —
!
10
6.5
1*
1 .45
8.76
11
1.02 7
.10
1
7.1
l #
4.48
3.77
5 f
!.70
9
4.40 1
i
10
7.2
5
1.82
6.59
2
3.03
8
.76
i
15
7.3
I*
4.73
3.30
7 1
.18
2
4.44 ! 5 5
.27
1
7.4
1«
1.34
1.29
10 1
.50
4
1.33
10
■
7.5
1"
3.49
2.03
11,
.95
3
2 . 70 !
5
true introduction period not known, value set equal tc
1
 27
TABLE 5: ACTUAL LIFE CYCLE STAGE AND GROWTH RATE OF PRICE ELASTICITY
G R
Product
W T H
g
MATURITY
Product g
D E C L
Product
. I N E
g
p
H
A
R
M
A
1.5
18.90
1.3
11.72
1.1
2.90
1.7
15.14
1.4
5.93
3.3
3.50
2.1
10.39
2.2
9.41
4.5
1.14
2.5
12.07
2.3
4.35
4.8
4.71
3.6
12.95
4.6
.66
4.1
12.43
4.2
19.89
mean
14.53
6.15
3.06
D G
E E
T N
E T
R S
5.4
14.28
5.1
5.50
5.5
4.75
6.4
3.34
6.1
1.17
6.5
8.76
7.2
6.59
7.1
3.77
7.3
3.30
7.4
1.24
7.5
2.03
mean
5.49
 .96
5.60
 28 
FIGURE 1: PRODUCT LIFE CYCLES
A
ZOOO 
1000 
Al: a =300, a ? =1.3, a ? =.05
Al: a.=1000, a 2 =.75, a 3 =.05
A2: a^lOO, a 2 =1.8, a 3 =.l
A2: aj=2000, a 2 =.l, a 3 =.l
i k i i i » i ' f i i » £
1 5 10
29 
FIGURE 2: PRICE RESPONSE FUNCTION AND PRICE ELASTICITY
"l.ti
e.
, i
,t
"V
\ •
A Kt l ;
4
3
2
1
it
•
•
•
•
•
• \
*—fr~
.7 .8 .9 1
1.1 1.2 1.3
'i,t
30
FIGURE 3: EXAMPLES OF PRICE ELASTICITY DEVELOPMENTS
Product
31
FIGURE 4: DISTRIBUTIONS OF THE MEDIANS OF PRICE ELASTICITY
i
40
i
i i
30'
20«
•— — i i
i
10
•
1 n >■ ■ 1
4=
.5 1 1.5 2
«? <
median
.44
median
1.88
pharmaceuticals
detergents
32
FIGURE 5: EXAMPLES OF CLASSIFICATIONS INTO LI
FE CYCLE STAGES
PRODUCT 1.1
PRODUCT 4.6
PRODUCT 5.5
Introduction „ . Growth ,„ =
IV
ty IV == Decline
 33 
FIGURE 6: AVERAGE PRICE ELASTICITIES AT DIFFERENT STAGES OF THE PLC
= e
l.t 1
c
i,t
DETERGENTS
INTRO GROWTH MATURITY
DUCT I ON
DECLINE
PHARMACEUTICALS
staae of PLC
34
FIGURE 7: OPTIMAL AND ACTUAL PRICING STRATEGY OF PRODUCT 4.2
p i)t .80
.70 
.60
.50
optimal
actual
pricing strategy
competitive price
i » i * i i i i i i
1 5 10
t (quarters)
35 
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JYOi
SEP 27 1991
ACME
BOOKBINDING CO., INC.
SEP 6 1983
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CHARLESTOWN, MASS.
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