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Mathematical Preliminaries
The
lognormal distribution
was used extensively in this study. A stochastic
variable X is said to be lognormally distributed (Λ) if its logarithm is
normally distributed, or using mathematical language, X ε Λ(μ, σ) if and
only if log X ε N(μ, σ). Here,
μ
is the average and
σ
is the standard
deviation. Lognormal distributions appear in a broad range of situations in
geology, biology, ecology, economics, and reliability theory (for surveys,
see Aitchison and Brown (1957) and Crow and Shimizu (eds.) (1988)).
A simple argument for this ubiquitous appearance of the lognormal
distribution is based on the central limit theorem; because sums of
stochastic variables under general conditions will converge to the normal
distribution, variables that grow with stochastically distributed growth rates
should be expected to converge to the lognormal distribution. Why incomes
or–as in the present study–prices in procurement operations should be
lognormally distributed is not obvious, but a possible justification is that the
price of a good or a service is generated via a series of mark-ups defined as
a percentage of the entry price. Fabiani et al. (2005) show that mark-up
pricing is the dominant method of pricing in the EU. In that way, the final
price becomes the product of a number of stochastically distributed price
increases. Lognormal distributions have been used previously in the
analysis of competitive bidding (see e.g. Laffont et al. (1995) and Skitmore
et al. (2001)).
By definition, a lognormally distributed stochastic variable has the
frequency function
f(x; μ, σ) = {exp[(ln(x) – μ)
2
/2σ
2
]}/ σx√2π, x > 0.
The following values can be derived:
average = exp(μ + σ
2
/2)
variance = exp(2μ + σ
2
) [exp(σ
2
) – 1)]
coefficient of variation = [exp(σ
2
) – 1)]
½
.
In many situations, including procurement, the minimal value or threshold
value is not zero but some positive value
τ
. This simply translates the
whole distribution
τ
units to the right. The average will increase by
τ
units,
whereas the standard deviation remains unchanged.
A population of suppliers is assumed to be available when public
procurement is announced. The ensemble of producers can be
characterized by a supply curve or distribution function F(.), showing how
the supply level varies with price. A number of these potential suppliers,
n
,
