In what follows, we consider what happens when different probability
distributions are used to model the behavior of the demand, namely
the lognormal, normal and triangular distributions. To enable comparisons,
we will suppose that the demand mean is always 
and the
unit prices 
and 
(cost to price ratio 
).
Among many distributions that can be used to model the behavior of
a positive quantity, the lognormal is the right one when the phenomenon
is the result of many independent random (multiplicative) factors.
fig:lognormal-densities shows the pdf of four lognormal
distributions, with common mean and various 
,
namely 
. The corresponding expected gains
(depending on 
) are given by fig:lognormal-gains.
When using a lognormal distribution only for convenience, one introduce
an extra dependence between variance and skewness since, for this
distribution, 
where 
is
defined as 
.
One can also try a Gaussian distribution. With the same mean ,
and the same four values of as before, fig:normal-densities
is obtained for the densities and fig:normal-gains for
the expected gains. It must be noticed that using a Gaussian model
would be sound if consumers were acting as additive processes whose
decisions were taken independently... but this is quite never the
case. Many decision criteria are the same for everybody, the welfare
of the general economy among them.
On the other hand, when using a Gaussian distribution only for convenience
(as it is often the case in the literature), it must be clear that
this choice implies a small variation coefficient (
)
in order to ensure that 
. In many situations,
this is quite unrealistic (using 
in this example
is questionable).
For the sake of comparison, let us also consider a triangular distribution,
defined by the bounds 
and the mode 
of the demand. Using 
as shape coefficient (see Appendix
A), together with the formerly
used values for 
and , the densities are given by fig:triangular-densities
and the expected gains 
by fig:triangular-gains.
|
[Densities.]
 [Expected gains and optimal locus.]
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[Densities.]
 [Expected gains and optimal locus.]
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[Densities.]
 [Expected gains and optimal locus.]
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In fig:lognormal-gains, 2(b) and 3(b),
we have drawn the loci of all the points 
obtained when 
varies. Obviously, these loci are all starting
from 
and their intersection with
a given curve gives exactly the extremal point of that curve. The
locus relative to the lognormal distribution is a curve that tends
to 
when 
, while
the loci relative to the Gaussian and the triangular distributions
are straight lines bounded by the conditions
for the normal one, and 
for the triangular
one (due to the choice of 
, cf. Appendix A).
For increasing values of , we see that the optimal order
shifts more and more towards left and the mean cease to be
a useful guess for .
If we now consider products with a lower cost to price ratio, for
example 
obtained by lowering cost to 
and keeping
, the superposition of all the profit curves corresponding
to the former considered distributions leads to fig:trois-gains-droite.
This time, the loci are shifting to the right when increase.
As before, the loci relative to the triangular and the normal laws
are straight lines bounded by a condition over . On the other
hand, the locus relative to the lognormal law must reach the origin
when : beyond a given value of ,
this locus stops shifting to the right, and undertakes a shift to
the left.
).
.