Munich Personal RePEc Archive
Evolutionary Model of Non-Durable
Markets
Kaldasch, Joachim
EBC Hochschule Berlin
27 September 2011
Online at https://mpra.ub.uni-muenchen.de/33743/
MPRA Paper No. 33743, posted 27 Sep 2011 12:41 UTC
Evolutionary Model of Non-Durable Markets
Joachim Kaldasch
EBC Hochschule Berlin, Alexanderplatz 1, 10178 Berlin, Germany
(email:
joachim.kaldasch@international-business-school.de)
Abstract
Presented is an evolutionary model of consumer non-durable markets, which is an extension
of a previously published paper on consumer durables. The model suggests that the
repurchase process is governed by preferential growth. Applying statistical methods it can be
shown that in a competitive market the mean price declines according to an exponential law
towards a natural price, while the corresponding price distribution is approximately given by a
Laplace distribution for independent price decisions of the manufacturers. The sales of
individual brands are determined by a replicator dynamics. As a consequence the size
distribution of business units is a lognormal distribution, while the growth rates are also given
by a Laplace distribution. Moreover products with a higher fitness replace those with a lower
fitness according to a logistic law.
Most remarkable is the prediction that the price distribution becomes unstable at market
clearing, which is in striking difference to the Walrasian picture in standard microeconomics.
The reason for this statement is that competition between products exists only if there is an
excess supply, causing a decreasing mean price. When, for example by significant events,
demand increases or is equal to supply, competition breaks down and the price exhibits a
jump. When this supply shortage is accompanied with an arbitrage for traders, it may even
evolve into a speculative bubble. Neglecting the impact of speculation here, the evolutionary
model can be linked to a stochastic jump-diffusion model.
Keywords: non-durables, evolutionary economics, economic growth, price distribution,
Laplace distribution, replicator equation, firm growth, growth rate distribution, competition,
jump-diffusion Model
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2
1. Introduction
The evolutionary model of non-durable markets presented here is an extension of a
previously published approach to consumer durable goods [1-3]. The key idea is that similar
to durable goods, also non-durables must be governed by the evolutionary Variation-
Selection-Reproduction (VSR) mechanism [4, 5]. The VSR-mechanism can be understood as
follows:
In an anonymous, competitive, non-durable market manufacturers offer the same good
with slightly different product features. We want to denote the product variants as brands. We
consider firms as consisting of a number of business units producing the corresponding brand.
The features are related on the one hand to utility properties of a brand and on the other hand
to its product price. There are non-durable goods which are per definition standards (e.g.
electric current). In this case the brands differ only in the price.
A business unit can be treated as an input-output system, where the input is the
financial revenue and the output is its product supply. Together with the demand side of the
market the business units form a reproduction cycle (R). During the reproduction process,
however business units vary the features of the brands a bit, in particular the price (V). The
preferences of potential consumers determine the selective "environment" for the brands (S).
Those brands which correspond closest to consumer demand will be preferentially selected.
As a result the reproduction cycle is self-amplifying and characterized by a preferential
growth. Specifying the growth rate as the fitness of a brand we will derive below that the unit
sales are governed by a replicator dynamics. In evolutionary terms the fittest brands have an
evolutionary (competitive) advantage compared to other products. Since brands with a lower
fitness have to increase their fitness or disappear from the market, the overall fitness of the
good will increase over a long period of time. In other words, the model suggests that Adam
Smith "invisible hand" is the evolutionary VSR-mechanism.
Applying statistical methods a model is developed that describes qualitatively the
evolution of a non-durable consumer market over a long time period. This model is valid if
the product fitness, which is a function of the price, has a single maximum. This maximum is
determined by the willingness to pay a minimum price, denoted as the natural price. However,
in difference to durables, non-durables are often the subject of speculation. In this case the
condition may not be fulfilled, because speculators are interested in an increasing price
instead to pay the natural price. They can disturb the market considerably. In order to keep the
model simple, the impact of speculators on the market evolution is neglected here.
Within this limitation the time evolution of the sales of the individual brands can be
derived from a consideration of demand and supply flows. Also derived is the size and growth
rate distribution of the business units (products), and the price distribution and mean price
evolution of the brands. Based on empirical investigations, the price evolution for non-
durables is usually modelled by stochastic differential equations. In this paper it is shown that
the evolutionary model can be mapped qualitatively to a standard jump-diffusion spot price
model.
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2. The Model
2.1. The Non-Durable Market
We start with a consideration of the (static) non-durable market.
The supply side
The supply side is determined by a number of different variants of the non-durable,
denoted as brands, having similar utility properties. They are produced and distributed by
manufacturers, while each brand is assigned to a business unit. We want to indicate the
brands (and the corresponding business units) with index i, while N>1 is its total number.
(We exclude here a monopoly market). The absolute number of units of the i-th brand sold per
unit time is denoted Y
i
, while S
i
indicates the number of supplied units.
The real price
i
of the i-th brand is the nominal price scaled by the mean income. The
mean price is determined by:
)()(
)(
1
)(
1
ttY
tY
t
i
N
i
i
t
(1)
while the brackets indicate the average over the sold units, while total unit sales and total
supply flow can be obtained from:
N
i
it
N
i
it
SSYY
11
;
(2)
The demand side
The demand side can be characterized by an ensemble of agents who are interested in
purchasing the non-durable. As for all physical goods the purchase process can be separated
into first purchase and repurchase of the good, while repurchase separates into replacement
and multiple purchase. In order to understand the first purchase process we have to emphasize
that some non-durables are coupled to a complementary durable good (called system
coupling). So for example energy is necessary for the application of engines in durable goods
(e.g. cars) and therefore first purchase of the corresponding nondurable (e.g. fuel) depends on
the diffusion of the durable good. We want to denote the number of agents, who have adopted
the durable good as N(t) and the total number of interested agents as the market potential M.
For inexpensive non-durables products, which have no system coupling (e.g. food) the
adoption process (N(t)) is limited by the information spreading process, which can be well
described by Bass diffusion [1,6,7]. However, the mean lifetime of the non-durable good is
short (usually in the order of less than a couple of weeks). Therefore correlations in the unit
sales caused by the replacement of the good will, in difference to durable goods, disappear
fast. Therefore repurchase can be treated as an effective multiple purchase. Because the first
purchase process evolves much slower than the repurchase process, we assume:
i) The repurchase of a non-durable follows the adoption process immediately.
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In this case we can define a total demand rate as:
)()()( tNtmqtD
(3)
where q>0 is a price dependent multiple purchase rate. It determines the mean number
of purchase decisions per adopter per unit time. The demand rate is proportional to the
number of adopters, N(t). Note that the demand of the non-durable is determined by the mean
amount, per purchase decision m. Usually non-durables can be purchase in single units, then
m=1 unit. But if this is not the case (e.g. fuel) the value of m has to be modified by a
corresponding average (e.g. mean tank content).
In order to establish a theory with continuous variables, we scale extensive variables
by the market potential M, which is a large figure. The corresponding density becomes:
)()()( tntmqtd
(4)
where n(t)=N(t)/M . The corresponding purchase and supply flow densities are determined
by:
M
S
s
M
Y
y
i
i
i
i
;
(5)
The key idea to describe the demand rate is the following assumption:
ii) It exists a minimum mean price
m
>0, at which the demand rate has a maximum.
Following Adam Smith we want to denote this minimum price as the natural price. This
assumption reflects the "law" of diminishing marginal utility of a non-durable (Gossen's law).
In other words, the willingness to pay has a maximum at
m
and hence the demand rate in
Eq.(4) has a maximum. Sufficiently close to
m
the total demand rate can be expanded as a
function of the mean price as:
2
mm
dd
(6)
where d
m
is a maximum demand. For some non-durables d
m
exhibits periodic variations (e.g.
electricity). Note that as mentioned above the impact of speculation is neglected.
2.2. The Market Dynamics
The market dynamics is determined by both, the supply and demand side.
