The author proposes a behavioural approach and not an econometric one to assess air passenger demand. He makes a very innovative use of revenue distributions for estimating the potential number of travellers within a population. It is based on a very limited set of assumptions. It was initially developed during the author's time in Airbus before having a larger utilisation (Aéroports de Paris, ADPi, ACI World, Bombardier, S aéro experts Inc, ...)

1. Daniel SALLIER 1
LONG-TERM DEMAND FORECAST: THE KENZA APPROACH
By Daniel SALLIER
Aéroports de Paris
Bât. 530 – Zone Orlytech
9, Allée Hélène Boucher
Orly Sud 103
94396 Orly Aérogare cedex
France
Telephone: +33 6 82 84 12 56
daniel.sallier@adp.fr

2. Daniel SALLIER 2
ABSTRACT
In this contribution I introduce a non-econometric method for demand forecasting. This method is
called the Kenza approach. It has been initially used by Airbus Industrie in the late 90s and
represents, today, the primary demand short, medium and (very) long term forecasting tool of
Aéroports de Paris (CDG & Orly). Because of very long lead-time developments and even longer
life cycles of their products/infrastructures, both aircraft and engine manufacturers, airports and
civil aviation authorities, if not the financial community, do have very strong methodological
requirements for fully supported long term forecasting techniques. One of the main assets of the
Kenza approach is to provide a better assessment and understanding on how the demand elasticity is
likely to evolve in the future.
To start with the empirical concepts leading to the 1st
Kenza law of demand will be exposed and
discussed. Then special attention will be paid to demand elasticity properties as they result from the
Kenza law of demand. To finish with the US domestic market will be used as a benchmark basis for
comparing different long-term forecasting models: the Kenza approach and more classical
econometrical ones.
Keywords:
Demand, Forecast, Modelling, Elasticity, Air Traffic, Passengers, long-term, consumer behaviour,
income distribution

3. Daniel SALLIER 3
1. INTRODUCTION
One of the dominant features of the aircraft manufacturers such as Airbus, Boeing, Canadair,
Embraer … is to deliver very long lasting products. For instance Northwest's DC9 fleet, a Mc
Donald Douglas aircraft, is about 30 years old and this is not even an exception.
Designing and manufacturing a new aircraft type is a very expensive and risky adventure. It
requires an investment of which the order of magnitude is US$ 8 up to 12 billion depending on the
aircraft type. In addition, between the date of the initial discussions about the opportunity of
developing a new type and the aircraft entry into service, 10 years may pass and it represents a very
minimum1
.
So the aircraft manufacturers require a (very) long time for making up their mind on either or not
developing a new aircraft which will have a very long product life expectancy, assumed it is a
"success" and which they will manufacture on a rather modest scale over a very long period of time;
Airbus A320 entry into service took place in March 1988. In March 2008, 20 years later, 1,881
units "only" of this very successful aircraft have been delivered to the customers; production figures
which cannot compare by any way with those of the car industry.
These considerations explain the very need this industry has for long term (15 to 20 years) and even
very long term (25 years and more) passenger demand forecast; bearing in mind that long and very
long term passenger traffic evolutions directly affect the overall project profitability and the
adequacy of the product design to the market requirements.
Engines manufacturers, airports, civil aviation authorities, together with the aircraft manufacturers
share exactly the same concern of having to deal with slow decision making process, highly
expensive manufacturing processes, without even mentioning the very long life expectancy of the
products/infrastructures they deliver2
. Because of the very long life expectancy, bankers specialised
in aeronautic asset financing, as well as credit export organisations need to have a view on the
future residual value of the aircraft type they are/will financing or backing which is, of course,
dependent on the aircraft world fleet demographic pyramid and its future deformations.
Econometrics3
provides the primary quantitative and qualitative demand forecasting tools used by
the experts worldwide4
, bearing in mind that producing a demand forecast is the inescapable
precondition for any subsequent studies on project technical feasibility and profitability.
Nevertheless using econometric models for long term forecasting is far more a "better than nothing"
1
1989 can be considered as the date of the preliminary discussions on Airbus A380 of which the first delivery of a
commercial aircraft to Singapore Airlines took place in 2007: 18 years!
2
This is not the case for airlines which are far more flexible and can change/adapt their product, their fleet structure,
their network, their business model, their network, etc. in far less that 5 years. A clear illustration is that most
airlines strategic planning is done on a time scale of 5 years maximum. Another illustration is, for instance, Air
France deciding to move from point to point network operation to hub-and-spoke ones in 1994 and having CDG
being its fully operational hub in 1996 (2 years).
3
"Econometrics" exclusively refers here at the use of statistical techniques in economics. It does not refer to an
extended definition which would be the use of (any) mathematical technique for the same purpose.
4
This is the case, for instance of, Airbus, Boeing, Rolls Royce, ICAO, ACI, Eurocontrol, French Civil Aviation
Authority (DGAC), Air France, major part of the US airlines, major part of the airports worldwide ... and we suspect
the list to be far, far more extensive.

4. Daniel SALLIER 4
approach than a fully satisfying solution, and having everybody using the same set of techniques,
even worldwide, does not make these techniques more accurate or more relevant5
.
They are 2 main reasons which sustain this "better than nothing" assessment:
1/ There is an "empirical" rule which demands any "classical" built forecasting model to be
based on times series which are twice as long as the forecasting horizon: i.e. 10 years long
time series for a 5 years ahead forecast, 40 years long time series for a 20 years forecast6
. It
means that long term forecast is fully depend on the actual availability of very long times
series and to the homogeneity of these historical data7
.
In addition, it is almost impossible to enjoy "clean" time series over (very) long period of
time. Most of the time a country is faced with events such as a terrorist attack, a paranoid
government, an oil crisis, an economical recession, a change in the market regulation, etc...
which temporally or definitively alter consumer behaviours, which shorten the number of
homogeneous data available to work with;
2/ In addition to this 1st
issue on historical data availability we should consider another related
issue which is that any econometric model extrapolates over the future the (consumer)
behaviour which has been implicitly measured over the past. Making such an implicit
extrapolation is likely to be very acceptable for short and even medium term forecasts further
to the very high inertia of social and economical behaviours of human populations. Our point
is that making such a "behaviour steadiness" assumption for long term forecasting is far more
questionable. Isn't product life cycle a phenomenon which reminds us how dubious it is to
extrapolate a market evolution out of its early/recent evolutions?
So it is "better than nothing" because both conditions are generally met: too short and "dirty" time
series and no or badly assessed future behaviour change which would, normally invalidate the use
of econometric models for long term forecasting purposes. Part of the modelling uncertainty is
assumed to be overcame by either "educated guess" post-processing of the models outputs, or the
use of the low-medium-high case approach or, more seriously, by extensive model sensitivity
analysis.
These are the main reasons why we developed the empirical Kenza approach which is exposed in
this paper. Our driving idea for doing so is that if it is possible to develop some kind of behavioural
modelling, the behaviour part of the model can add up to the missing information in time series too
5
"Vox populi, vox Dei" does not apply in science and technical issues.
6
A way to look at this question of historical length of time series could be to relate it to very similar questions raised
by short-term fast Fourier transforms. The set of historical data can be considered as resulting from the use of a time
window on the sampled signal (i.e.: monthly data). Doing so cannot but introduce a spectrum deformation in the
frequency space with regard to the "signal" spectrum which would result from a "classical" Fourier transform.
Medium and long term forecast is much more "influenced" by the medium and low frequencies of the signal
spectrum than by its high frequency components. The time window should present a time width which is compatible
with an "acceptable" alteration of the low and medium frequency components of the signal spectrum. A paper is left
to written which would address this issue.
7
With the only exception of the USA, it is almost impossible to "enjoy" 40 years long and more, homogeneous (same
definition over the time) time series for the GDP, CPI, exchange rates, etc. anywhere else in the world.

5. Daniel SALLIER 5
short with regard to the forecasting horizon. In addition and because of their very nature,
behavioural modelling can take into considerations (consumer) behaviour changes8
more easily.
The behavioural modelling we detail in this paper is mostly based on the assumption that a causal
link (not a correlation) can be established between distribution of income and passenger demand.
Assuming the existence of a link between household/individual income and demand has been
already done for decades. The first and obvious link is to consider household, individual or
economical agent revenue as the total amount of money he can spend. Isn't (anticipated) utility
concept in this category of link since it is at the very base of economical theories on consumption
choice between different goods for a given amount of money9
? Isn't "discretionary income", very
often mislaid with disposable income, in this category of conceptual link since it represents what is
left for current spending once taxes and normal expenses (such as rent or mortgage, utilities,
insurance, medical, transportation, property maintenance, child support, inflation, food, ...) have
been subtracted from the personal/household income?10
Vilfredo PARETO interest for the
distribution of (household) income within a population which gave birth to the Pareto distribution
came from a slightly different economical interest which had much more to do with tax policy and
household wealth11
. It looks like studies on income distributions should be mostly found in
academic departments of sociology12
. The only authors we found and who have been trying to
study/establish a link between consumer demand and income distribution are those working on
Werner HILDENBRAND's market demand theory13
, but their interest is much more in the
demonstration of the general equilibrium theory than in trying to model customers demand for one
specific good such as air travels. This is one of the reasons why we are faced with a rather thin
bibliography base to work with.
The initial application of the first Kenza law of demand took place in 1995 during our time in
Airbus Industrie and was used as a marketing support of various aircraft sale campaigns all over the
world. In 2002, we dedicated additional R&D work to the first Kenza law of demand for it to
provide the theoretical background of the short, medium and long term demand forecasting tool of
Paris Airports Authority (Aéroports de Paris).
The first Kenza law of demand which will be detailed hereafter is for:
8
Another way to illustrate the idea is, for instance, the Newton law of motion. The Newton law can be considered as
a behavioural model which allows very precise and accurate far ahead anticipations on the trajectory of any stellar
object such as a comet, for instance, anticipations which can be assessed out of a set of measures which would be far
insufficient for achieving the same result with statistical approaches.
9
For instance the utility concept is developed by Léon WALRAS "Éléments d'économie politique pure ou théorie de
la richesse sociale", Economica, Paris, 1988
10
Kathleen SHORT, "Alternative Poverty Measures in the Survey of Income and Program Participation: 1996", U.S.
Census Bureau, January 3, 2003".
11
Vilfredo PARETO, "Cours d'Economie Politique", reproduit en 1964, in Vilfredo Pareto, Œuvres Complètes,
Genève : Librairie Droz
12
Over a total of 253 papers listed by the Michigan University internet site having "Household income" as a keyword,
58% are about socio-economical topics such income/wealth inequality, 33% refer to macro/sector economical issues
such as tax policy or wealth accumulation for instance. Household consumption studies represents 9% of the total
number of titles listed with roughly similar importance (in terms of number of publications) for housing issues and
methodological issues (i.e. household consumption utility function). There is only one title over 253 about
household demand.
13
Werner HILDENBRAND in his work on market demand ("Market Demand – Theory and Empirical Evidence",
Princeton University Press 1994) introduces explicit references to household income distributions. The very
difference with W. HILDENBRAND theory is that the subject of interest is the general equilibrium theory and he
looks at the dispersion of the basket of goods household consume according to the household income evolution.

6. Daniel SALLIER 6
1/ modelling annual passenger demand. It is not designed for modelling quarterly, monthly,
weekly, daily, ... traffic and for taking into consideration traffic seasonality for instance;
2/ modelling generic traffic flows which can be hardly substituted by other traffic flows. The US
domestic air market can be considered as a generic market while the leisure market of Tunisian
destinations from France should be considered as a specific market which can be substituted by
similar destinations at any time such as Moroccan destinations for instance. Specific markets
can be modelled by the 2nd
Kenza law of demand which is not exposed in this paper;
3/ modelling origin/destination flows which means, for instance, that modelling traffic flows
between the USA and Western Europe would require, at least14
, two different models to be
developed, one for the passengers living in the USA and a second one for those living in
Western Europe.
To start with, we will detail the empirical approach which lead to the first Kenza law of consumer
demand. We will use the US domestic air market to illustrate the ability of the 1st Kenza law of
demand to provide both a good estimate of traffic data and of a good long forecasting model of air
traffic/demand. Then we will pay a specific attention to demand elasticity as it results from the first
Kenza law of demand, bearing in mind that "educated guess"15
tells us that market maturity should
necessarily induces a declining demand elasticity to GDP, for instance. But it is still pretty difficult
to provide statistical/theoretical proves supporting this phenomenon16
. To finish with, before
concluding, we will use again the US domestic air market as a long term forecasting methods
benchmark test.
2. THE FIRST KENZA LAW OF DEMAND
The very idea at the origin of the empirical approach which is detailed hereafter was to try assessing
the very reason, the structural cause why people where flying. This question took place out of any
actual data considerations. The only answer we came out with was that people are flying because
they can afford it17
. We refined this broad initial "explanation" by considering that the very reason
why an individual buys a ticket or an all-inclusive tour is because:
14
Transfer flows such as Asia to the USA (East Cost) via Western Europe would require additional modeling efforts.
15
That is how the French Civil Aviation Authority decided to consider a 1.5 traffic elasticity to GDP for its long term
forecast. The 1.5 figure is a typical example of educated guess no supported by any statistical analysis on the traffic
historical evolution.
16
The issue with demand elasticity is that it is not an "absolute" economical quantity the way pressure or temperature
are physical quantities. Temperature can be measured in different units (Celsius, Kelvin, Fahrenheit, ...) but keeps
being the same physical phenomenon, while demand elasticity value and time evolution is fully dependant on the
explicit/implicit demand model which is used by the "observer". For instance, a log-linear model provides a constant
demand elasticity to GDP while a linear model provides a declining one over the time. Unfortunately the same set of
historical data can very often be fitted very accurately with the two different types of model resulting in two very
different estimate of the demand elasticity to price or GDP for instance.
17
This approach can sound a bit weird since the "normal" way to proceed is to look at data sets and try to figure out
what can link them together. In fact, it is not the "normal" way to proceed in physics or in system dynamics, for
instance, for which the normal way is trying to figure out the (structural) causes at work behind phenomenon:
develop a conceptual model first of which the validity directly result from its capacity to "explain" actual
measurements.

7. Daniel SALLIER 7
1/ he feels like buying it, or have to buy it. Some would relate this "feeling" or this "need" to the
concepts of value or utility;
2/ he can afford it;
3/ he is sensitive to the relative price with regard to his own revenue18
.
It results from this idea that for a given average price of the air ticket or the inclusive-tour there is
corresponding threshold of (annual) individual income beyond which, generally speaking, potential
customers should be found: the elected population.
This elected population derives directly from the threshold of individual income and the income
distribution within the population. Of course, actual customers represent only part of this elected
population and each of them buys only a given amount of the good/service units.
This is illustrated in the following chart:
Figure 1: the empirical principle of the demand law
In this chart, the red thick line represents the cumulative distribution of individual income or, more
precisely, the total number of inhabitants of which the annual income is greater or equal to the
corresponding amount on the X axis.
Reformulation of the initial principle
We reformulate the former chart by considering percentage of the total population on the Y-axis
instead of total population itself and by considering relative price/income to a price/income
18
The individual income is defined as Keynes' household (annual) monetary income divided by the number of
household members. We do not take into consideration the age of the household members (i.e. adult, baby, child) in
order to "correct" the corresponding individual income. We do not consider the disposable revenue either and even
less the discretionary revenue.

8. Daniel SALLIER 8
normalisation variable such as the GDP per capita or the average/median individual income on the
X-axis instead of price/revenue stated in monetary values such as in US$, Euro, Yen, etc..
The former chart is "redesigned" as follows:
Figure 2 : reformulated principles of the demand law
In the former chart the red thick line  
*
n
F r represents the 1-complement of the cumulative
distribution of normalised individual income – Kenza distribution – it is to say the percentage of the
population of which the annual income is greater or equal to which is equal to the individual
annual revenue divided by the normalisation variable
n
r
r r which might be equal to the GDP per
capita or the average/median annual individual revenue, …
Kenza distributions
Kenza distributions and their properties would deserve a full specific paper.
Kenza distributions are determined out of population surveys or micro-census on household
incomes such as the ones produced by the US Census Bureau for the United States19
or Banca
d'Italia for Italy20
. Most of the time the data come in a double entries table providing the number of
households which belong both to an household categories (no. of household members) and an
annual income categories.
The following table illustrates the kind of data provided by the US Census Bureau
19
US census Bureau "CPS - Current Population Survey" which can be found on its internet site.
20
Banca d'Italia "SURVEY OF HOUSEHOLD INCOME AND WEALTH" which is a micro-census which can be
found on its internet site

9. Daniel SALLIER 9
One of the major empirical characteristics of the Kenza distributions is to be very steady over the
time which is illustrated in the following chart for the US population between the fiscal years 1994
and 2004.

10. Daniel SALLIER 10
Figure 3: US Kenza distribution evolution
We should point out that steadiness of the Kenza distribution of a country does not translate into
any sort of steadiness of the number, the size and the economical status of the households belonging
to this population: it is a macro population property, it is not a micro one. The US case is very far
from being unique21
. We should point out that this empirical steadiness over the time, no
21
In numerous papers and books Werner Hildenbrand made explicit statements on the stability/steadiness of income
distributions mostly based on the British case. This is, for instance, the case of an article he co-signed with Alois
KNEIP and Klaus J. UTIKAL entitled: "Une Analyse non Paramétrique des Distributions du Revenu et des
Caractéristiques des Ménages" in Revue de Statistique Appliquée, volume 47, No. 3 (1999), p. 39-56.
Very logically the authors based their study on the relative household income, using the average household income
as a normalisation value (p. 40). It came out of their statistical analysis that the relative income distribution is quite
stable without being strictly or even reasonably invariant (p.48):
"Lorsque nous avons considéré l'évolution des densités du revenu dans le temps, nous avons déjà remarqué plus
haut que la distribution du revenu nominal se modifiait rapidement dans le temps. La transformation du revenu
nominal en un revenu relatif conduit à des distributions beaucoup plus « invariante ». Néanmoins, ce caractère
d'invariance est loin d'être parfait."
In another paper "Handout for the Seminar by Werner Hildenbrand on Aggregation under Structural Stability",
Werner Hildenbrand wrote:
"(i) The change over time in distr(y; a / Ht)
We first consider the income distribution distr(yht j h 2 Ht). In pure theory the shape and the change over time in
these distributions can be arbitrary. The observed income distributions of actual economies however evolve over
time in a surprisingly “structurally stable” way. Income distributions have been studied extensively in the literature,

11. Daniel SALLIER 11
economical sociological theory seem to have explained up to now is very central in the long-term
capabilities of the Kenza approach.
Price and revenue normalisation
In fact, the average (annual) revenue per capita should be favoured as the normalisation quantity for
other monetary stated data. The main reason for doing so has to do with the Kenza distribution
steadiness over the time. But, in fact other data, such as the GDP per capita can be used as long as
they very strongly correlate with the average revenue per capita22
.
The use of normalised price/revenues values allows us to get rid of the everlasting issue of
modelling the demand function on real or current monetary values. We get exactly the same
normalised price/revenue calculated out of current or real monetary values as long as it is the same
inflation rate or CPI which is used for both the price/revenue data and the normalisation variable23
.
In addition, using price stated in normalised value allows us to get rid of the bias which results from
the evolution of the exchange rate between 2 different currencies which is a recurrent issue of
demand modelling in air transport since we have to deal with international flows and customers
living in different countries.
Demand equation
The chart number 2 can be translated into the following equation which represents the demand
function:
starting with Pareto (1897) and Bowley (1915). For recent references see Atkinson and Bourguignon (2000). The
empirical studies support the following".
22
The following chart represents the US GDP per capita as a function of the US average revenue per capita over the
1967-2005 period of time.
The better correlation, at least up to 2002 is provided with data stated in current value.
23
No mention made here of the problem raised by the fact that "real" GDP are now expressed in chained volume and
no more in real money.

12. Daniel SALLIER 12
 
n
p
K
F
K
K
P
D 



 2
*
2
,
1
1
,
1
Where:
r
p
pn  is the normalised average ticket or inclusive-tour sale
price, which is actual price divided by the normalisation
quantity at a given date t;
n
p
K 
2 is the threshold of normalised income;
 
n
p
K
F 
2
*
is the proportion of elected population relative to the total
population which can afford to buy the considered service
or the perishable good;
 
n
p
K
F
K 
 2
*
1
,
1 is the proportion relative to the total population of actual
customers. It results that 1
1
,
1 
K ;
 
n
p
K
F
K
P 

 2
*
1
,
1 is the actual number of consumers, P is the number of
inhabitants;
 
n
p
K
F
K
P
K
D 



 2
*
1
,
1
2
,
1 is the total number of service/good units actually
bought/sold; t is the demand while represents the
average number of units actually bought by each
consumer.
D 2
,
1
K
It should be pointed out that the demand function takes explicitly into considerations the ticket price
(average), the income change of the customers (normalisation value) and various customer
behaviours such as which explicitly refers to the reservation price
2
K 24
and 1,1
K which is similar to
a market penetration ratio25
.
This very general equation is simplified by assuming that ,
2
K 1,1
K and 1,2
K are constant over the
time, but differ from one set of populations to another. Such an assumption is equivalent to assess
that consumer behaviour is fully characteristic of a given population and not likely to change over
the time.
This is a very highly demanding assumption which translates a phenomenon of social percolation of
consumer habits. It is to say that individuals tend to adopt the qualitative and quantitative
consuming habits, the standards of life, of the social class they join as soon as there income
improvement allow them to do so26, 27, 28, 29
.
24
Reservation (reserve) price is a very common concept in micro-economy one can find defined, for instance, by Ian
Steedman (1987). "Reservation price and reservation demand," The New Palgrave: A Dictionary of Economics, v. 4,
pp. 158-59.
25
The actual market penetration would be calculated on the total population and not on the elected population only.
26
It is a model of consumer behaviour which has been proposed and detailed by James Stemble DUESENBERRY in
"Income, Saving and the Theory of Consumer Behavior" Harvard University Press – January 1949.
27
The following chart represents the logarithm of Paris Airports traffic since 1951 as a function of the logarithm of the
French GDP. During this period of time the traffic went from 1.04 M passengers in 1951 up to 83.01 million in
2009. Despite a complete change of its socio-demographic base and quite a number of crisis/events France and the

13. Daniel SALLIER 13
1,1
K and 1,2
K are concatenated into a single constant 1 1,1 1,2
K K K
  and the 1st
Kenza law of
demand can be rewritten as follows:
     
 
*
1 2 n
D t P t K F K p t
   
Kenza model calibration process
Kenza calibration process requires that K1 and K2 to be determines out of the sets of historical data.
Very classically we propose the minimisation of the sum of quadratic errors to be our calibration
criteria.
To start with, let us assume that we already know K2 value. The sum of quadratic errors e² is equal
to:

 
 
 




n
i
n
i
i p
K
F
K
P
D
e
1
2
,
2
*
1
2
 where
i index, refers to the date i in the set of data
2
K 1,1
world went through, the graph shows a quite impressive steadiness of the traffic/GDP link which supports (without
proving) the assumption made on , K and 1,2
K .
28
It echoes the metonymy concept W. HILDENBRAND has developed in its market demand theory. While we
assume here the existence of a percolation concept which is that people tend to adopt the consumption habits of the
social/economical class they enter with an increase of their income, the metonymy concept assumes that with an
increase of its revenue a household adopt the same dispersion of its basket of goods as the one of the
economical/social class of the households it is sharing the same level of revenue.
29
Ground transportation shows very steady behaviours too. Yacov ZAHAVI in "Travel Time Budget and Mobility in
Urban Areas" (Final Report 1974, Department Of Transportation) identified a very steady "auto travel time in any
urban area". Same analysis made by Andreas SCHÄFER and David G. VICTOR, 2000, in "The Future Mobility of
the World Population", Transportation Research A, Volume 34, issue 3: pp 171-205 : "On average a person spends
1.10 h per day travelling and devotes a predictable fraction of income to travel. We show that these time and
money budgets are stable over space and time and can be used for projecting future levels of mobility and transport
mode. The fixed travel money budget requires that mobility rises nearly in proportion with income. Covering
greater distances within the same fixed travel time budget requires that travelers shift to faster modes of transport"

14. Daniel SALLIER 14
i
D actual demand measured at the date i
 
 
i
n
i p
K
F
K
P ,
2
*
1 

 Kenza demand estimate at the date i
For this given value of K2, e² is minimised for
 
 
 







 n
i
n
i
n
i
n
i
i
p
K
F
P
p
K
F
P
D
K
1
2
,
2
*
1
,
2
*
1
K2 can be determined by using an iterative dichotomy approach to minimise e² value. It requires to
start with 3 values of K2 (K2,min, K2 and K2,max) so that:
e²(K2)<min(e²(K2,max), e²(K2,min))
Then 2 new values of K2 are tested:
2
min
,
2
2
1
,
2
K
K
K

 and
2
max
,
2
2
2
,
2
K
K
K

 on the 2 different
intervals [K2,min, K2] and [K2, K2,max] which minimise e². According to which interval the new value
of K2 is found, the former value of K2 is equal to K2,min (second interval) or K2,max (first interval).
And then the process can be iterated again up to finding a value of K2 which meets convergence
criteria.
This calculation algorithm, pretty easy to implement, proves to be convergent30
.
3. THE US DOMESTIC AIR MARKET FOR ILLUSTRATION
The US domestic air market together with the Japanese domestic market is among the only ones in
the world for which all the data required by the 1st
Kenza law of demand are publicly available and,
in addition, over a long period of time.
General context of the US domestic market
1979-1982 Deregulation of the US domestic air market. Since 1983 the US domestic market has
been fully regulated by a federal agency called the CAB (Civil Aviation Board) granting domestic
traffic rights to the airlines which included fares, weekly number of flights, available seats, etc...
As soon as 1970 the regulation system is considered as being far too rigid by all the actors or the air
transport community, cities, counties and passengers included. Starting 1975 a commission is
30
Using the minimisation of the sum of quadratic error as criteria for calibrating Kenza law parameters may sound a
bit confusing with regard to our claim of proposing a non econometric model since its is exactly the same kind of
mathematical techniques which is used for calibrating any econometric model. Is Kenza an econometric approach?
The answer is yes if, by econometrics, one considers as being econometrics the use of mathematical tools in
economics, but this is a quite a broad definition. If one considers that econometrics is the art of finding correlations
between economical quantities and of developing mathematical models which results from these correlations, Kenza
is definitely not an econometric approach while at the same time a Kenza estimate can correlate very closely with
the traffic it models. Once again, we are faced here with the everlasting question of the difference between causality
and correlations.

15. Daniel SALLIER 15
settled - the United States Senate Judiciary Committee - which starts auditing the different parties
with the explicit aim of deregulating the US domestic air market.
On October the 24th
, 1978, President Carter signed the deregulation act which assessed the end of
any operation restriction by December the 31st, 1981 at the latest and the end of fare regulation by
January the 1st
, 1983.
The period of time between January the 1st, 1979 and December the 31st, 1982 should be viewed as
a period of transition between two very different systems.
The following chart sums up the main events of the US domestic market since 1975:
Figure 4: US Domestic traffic evolution
The chart clearly illustrates how difficult it is to get long, homogeneous and "clean" time series. In
our present case, over a period of 34 years to which correspond 34 annual traffic data, only about 18
can be used (1981 until 1990, 1994 until 2000 and 2005 until 2008) for being uneventful periods.
Estimate of the US domestic market by Kenza modelling
The US domestic air market cannot be strictly considered as a consumer market since part of the
passengers are having business trips for which their tickets is paid by their company. Nevertheless,
companies tend to pay air tickets almost exclusively for their executives and technicians, far less
frequently for the cleaning persons or the clerks for instance. So company paid tickets seems to be
quite related to the professional status of employees which reflects in their revenue (even in

16. Daniel SALLIER 16
communist countries). Because of this very specificity of air transport, the Kenza approach can
apply on this market.
The following chart represents the Kenza estimate of the US domestic air traffic.
The linear regression between the actual traffic data and their Kenza estimate over the uneventful
periods 1981-1990, 1994-2000 and 2005-2008 results in a R²=0.975. It looks like the Kenza
approach delivers what it has been designed for: ability to provide long term forecast out of a
limited number of historical data.
4. DEMAND ELASTICITIES
Let us define the intrinsic elasticity of a Kenza distribution as:
   
 
0
*
*



n
n
n
n
n
K
dr
r
r
F
r
dF
r

This quantity is related to the Kenza distribution shape out of any direct considerations on demand
functions.

17. Daniel SALLIER 17
It can be proved that the intrinsic elasticity is completely independent of the normalisation quantity
which is used (i.e. GDP/capita, average individual income, median individual income, etc...) and,
thus, is fully characteristic of the population which is studied.
Assumed that the demand is modelled by the first Kenza law of demand
     
 
*
1 2 n
D t P t K F K p t
   
The demand elasticity to price p is equal to:
 
n
K
p p
K 
 2


and the demand elasticity to the normalisation quantity r (i.e. GDP per capita or average revenue
per capita) is equal to:
 
n
K
r
p
K 

 2


So demand elasticities directly derive from the Kenza intrinsic elasticity.
Analysis of actual Kenza distributions of several countries, such as the USA, shows quite large
section of an almost linear relationship between the intrinsic elasticity and the corresponding
elected population (sorted by decreasing level of income) The following chart represents the US
intrinsic elasticity as a function of the US elected population together with US domestic air market
price elasticity at different dates of its evolution:

18. Daniel SALLIER 18
Figure 5: US intrinsic elasticity & US domestic demand elasticity to price
So, the 1st
Kenza law of demand takes "naturally" into account the long term market rather linear
maturation process (demand elasticity to price decreasing in absolute value). It should be pointed
out that the market maturation process is not related to the market which is studied. It means that, in
our case, it has nothing to do with the US domestic air market itself. It exclusively results from the
way individual incomes are distributed with the considered population31
. The only phenomenon
which is market related is how fast the elected population grows or decreases over the time32
.
Beyond pure modelling and forecasting concerns, from a much more prospective point of view, a
good understanding of how the demand elasticity of a given economical sector is likely to evolve
can provide clues on how the sector can (re)structure itself in the future. It has to do with the sector
"trying to keep its ability" to make profits in a tougher competitive environment33, 34
.
31
One should bear in mind that Kenza distributions are pretty steady over the time, which means that the intrinsic
elasticity as a function of the elected population is pretty invariant too; an interesting property for long term
forecasting.
32
A high K2 market (low reservation price) is a market which is maturing faster than a low K2 market, but in both
cases the same level of elected population results in the same demand elasticity.
33
Let us define 1 C
V
V
M
T
  where VC = total variable cost of the sector and T= sector turnover. It can be proved
that if the demand elasticity to the average sale price
1
p
V
M
   then a decrease of the average unit sale price p
of the sector results in profit decrease of the overall sector. For instance, let us assume that the demand elasticity to
the average price is -1.2, then total variable costs of the sector should not exceed 17% of the sector turnover
otherwise decrease of the average unit sale price results in a profit decrease which demand costs to be cut down for
the sector to resume its former level of profit.

19. Daniel SALLIER 19
5. KENZA MODEL VERSUS MORE "CLASSICAL" ONES
Once again we will use the US air domestic market, but this time it will be the test bed of the Kenza
modelling of the traffic and of eight different econometrical models build on the following sets of
explanatory variables:
- GDP stated in current and in real US dollars, airlines average revenue per passenger stated in
current and real US dollars and US population;
- GDP/capita stated in current and in real US dollars and airlines average revenue per passenger
stated in current and real US dollars.
Each set of explanatory is fitted with a log-linear and a linear model.
We have selected the 1982-1990 period of time as a reference to calibrate the different models for
the 1982-1990 period is a rather "peaceful" period of air transportation in the USA. 1982 comes just
after the US sky deregulation in 1979/1980 and 1990 is on the eve of Gulf War the 1st
(1991) which
seriously altered the US and World traffic dynamic.
Figure 6: US domestic air traffic estimates
34
The structural ability of a sector (i.e. all the US domestic airlines together) to make profits or losses does not tell us
anything about the ability of a specific airline (i.e. Southwest or JetBlue) to keep making profits.

20. Daniel SALLIER 20
The 9 demand models demonstrate excellent calibration characteristics over the same common
1982-1990 period of time which cannot allow us to identify the best one(s) among them. A
backward traffic forecast over the 1991-2009 period of time allows us to identify the Kenza model
as being the best candidates for delivering long term forecast and the linear model in real US
dollars, GDP, price & population based, as providing rather accurate 10 years ahead forecast. One
will notice that both Kenza and GDP, price & population based models give very close outputs for
short and mid-term forecast (up to 1996 with the exception of the log-linear model in current $).
GDP/capita based models do not provide good forecasting models even for short and medium term
forecasting.
Traffic estimates given by each of the 9 models are shown in the next array. R² value for the Kenza
model is the R² of the linear estimate between actual traffic data and corresponding Kenza estimate
over the calibration period (1982-1990). It compares with other models R².
One of the problems any forecaster is left with is to identify a good forecasting model out of a
bundle of models which present very close estimate qualities over the calibration period. To make a
long story short, one would try not to select the brown model and the dark blue ones of "Figure 6:
US domestic air traffic estimates" while both models present almost perfect calibration
characteristics over the 1982-1990 calibration period. In addition backward forecast in order to test

21. Daniel SALLIER 21
the forecasting quality of a model is a game one can hardly play most of the time for one should use
the longest possible historical data for accurate calibration purposes which leaves the forecaster
with no or very little data for backward forecasting tests. The pending question is: is my model
based on a "local mathematical/statistical coincidence" or does it reflects more structural
correlations or causalities? One way of addressing this question is to look at the model parameter
steadiness when changing the calibration period. The idea is that a demonstrated steadiness of a
model parameters is likely to characterise not only a "robust" model, but a model which is likely to
be "rooted in reality".
The following table shows the evolutions of the parameters of the 3 best forecasting models of our
benchmark. 1982-1990 is kept as the reference period for models calibration for the market
behaviour (airlines and passengers) has been rather steady. Other calibration period are chosen
between 1982 and 1990.
Within the frame of the very specific example of the US domestic market, the Kenza model gets the
best equation parameters stability criteria by far.
One of the main characteristics of the Kenza approach is to be based on the use of the Kenza
distribution which makes the direct (non linear) link between people revenue and the demand curve.
This is the reason why we introduce now the simplified Kenza function based on a local
linearization of the Kenza distribution of income over the calibration period:
 
   
B
p
A
P
b
p
K
a
K
P
D n
n 








 2
1 .
which can be written slightly differently: B
p
A
P
D
n 


which could be considered as a very simple econometric linear model to calibrate.

22. Daniel SALLIER 22
The difference between the full Kenza approach and the simplified one allows to measure the
"added value" of taking into account the Kenza distribution function which directly drives the
market penetration dynamic on the traffic (elected population evolution).
Figure 7: Kenza estimate of the US domestic traffic
6. CONCLUSION
The first Kenza law of demand looks like delivering what it has been designed for:
- an ability to provide rather accurate long-term forecast which is illustrated in the case of the US
domestic air market;
- A pretty robust model of which the low sensitivity of the parameters to changes in the
calibration period could be considered as an indirect confirmation of the "behavioural" nature
of the Kenza models. Furthermore, beyond the delivery of forecast outputs, it provides causes
and reasons why the traffic has evolved and would evolved this or that way in the past and in
the future. Population, price and GDP allow to position the threshold of income and its
evolutions and, doing so, give direct access to the elected population and its dynamics. Access
to the demographic, social and economical background of a market prove to be a valuable asset

23. Daniel SALLIER 23
mostly when presenting the outputs to a top management meeting. It proves far more
comfortable than coming with correlations!;
- an unanticipated ability at the time of the method development to provide additional valuable
pieces of information such as the intrinsic elasticity and how it directly drives the long term
evolution of demand elasticities and may provide clues on the market maturation process35
;
The issue with the approach which is exposed here is that one can raises the argument that this
approach can but work for it is based on the use of a S-curve (the Kenza distribution). And that the
modeller can deform this S curve to its very needs considering that playing with K2 allows to stretch
it horizontally along the X-axis and, so doing, allows the modeller to adjust the trend slope of its
Kenza estimate, while K1 allows to vertically stretch the same S curve, along the Y-axis, which will
allow to adjust the absolute value of the Kenza estimate to the actual data.
The S-curve argument can explain the reason why a Kenza model can provide good estimates for
historical data, no better estimates, by the way, than those given by more classical approaches. The
S-curve argument can explain the reason why a Kenza model presents short and medium term
forecasting capabilities which are time horizons for which the general trend of a "healthy" estimate
can be extrapolated for producing this type of forecast, a phenomenon which is illustrated in Figure
75 which represents both a Kenza model and its "trend" version the simplified Kenza model. The S-
curve argument cannot explain long-term forecasting capabilities of the Kenza model for the non
linear correction brought to the long-term demand estimate by the Kenza distribution cannot, of
course, be taken into consideration in the calibration process. Once again "Figure 7: Kenza estimate
of the US domestic traffic" provides an illustration of this fact.
In fact, the very "disturbing" issue with this approach is the very demanding assumption made on
K1 and K2 of being constant over the time. Of course we can find examples of very steady
economical/behaviour phenomenon over very long period of time and, of course, we quoted some
of them (note 21 and 26 to 29) to sustain our point. It simply proves that it is possible, it does not
really prove that it is true. This is the reason why we have conducted additional research works, yet
to be published, which seem to prove the case; Ks property would result from the aggregation to all
a population of a probabilistic model of individual consumption behaviour.
To finish with, we have developed and used the first Kenza law of demand for modelling air traffic
and passenger demand, but this approach far exceed the only topic of air transportation and can
address all sort of consumer demand issues.
7. REFERENCES
Chung R., Cockerham J., Donaldson D., Ellerd-Elliott S., Herring I., Hirani B., McColl H., Shome
Duesenberry J., "Income, Saving and the Theory of Consumer Behavior", Harvard University Press
(January 1949)
35
Is it a complete coincidence that, for instance, market deregulations such as the US, the Japanese or the Europeans
ones took place at the times when their demand elasticity to fares was close to -1?

24. Daniel SALLIER 24
Clementi F. and Gallegati M., "Pareto's Law of Income Distribution: Evidence for Germany, The
United Kingdom and the United States", New Economics Papers (NEP), NEP-MIC-2005-05-23
(Microeconomics)
Hildenbrand W., "Market Demand - Theory and Empirical Evidence", Princeton University Press
(1994)
Hildenbrand W., "Aggregation under Structural Stability", Handout for the Seminar
Hildenbrand W., Kneip A., Utikal K., "Une Analyse non Paramétrique des distributions du Revenu
et des caractéristiques des Ménages" Revue de Statistique Appliquée, tome 47, N° 3 (1999), p. 39-
56
Jerison M., "Demand Dispersion, Metonymy and Ideal Panel Data", Department of Economics,
SUNY, Albany
Keynes J., "The General Theory of Employment, Interest and Money", Palgrave Macmillan, 1st
edition (1936), Book II - "Chapter 6. The Definition of Income, Saving and Investment"
Kostanich D., Dippo C., "Current Population Survey Design and Methodology", U.S. Census
Bureau and Bureau of Labor Statistics - Technical Paper 63RV (March 2002)
Lewbel A., "Weren Hildenbrand's Market Demand", - Journal of Economic Literature, Vol. XXXII,
December 1994, pp 1832-1841
Pareto V., "Cours d'Economie Politique", in "Vilfredo Pareto, Œuvres Complètes", Genève :
Librairie Droz (1964)
Schäfer A. and Victor D. (2000), "The Future Mobility of the World Population", Transportation
Research A, Volume 34, issue 3: pp 171-205
Short K., "Alternative Poverty Measures in the Survey of Income and Program Participation: 1996",
U.S. Census Bureau (January 3, 2003)
Steedman I. (1987), "Reservation price and reservation demand," The New Palgrave: A Dictionary
of Economics, v. 4, pp. 158-59.
Stewart K. and Reed S. (1978-98), "Consumer Price Index Research Series Using Current
Methods", US Bureau of Labor Statistics - Monthly Labor Review (June 1999), p. 29
U.S. Census Bureau, Current Population Survey (CPS),
Internet site: http://www.census.gov/hhes/www/income/incomestats.html#cps
U.S. Census Bureau, Population Estimate,
Internet site: http://www.census.gov/popest/estbygeo.html
U.S. Census Bureau,Incompe - Historical Income Tables - people,
Internet site: http://www.census.gov/hhes/www/income/histinc/incpertoc.html
U.S. Department of Commerce, Bureau of Economic Analysis, "An Introduction to National
Economic Accounting", Methodology papers Series MP-1 Washington, DC: (GPO, March 1985)
Velupillai K., "Werner Hildenbrand and the general equilibrium theory – Will we have to rewrite
our textbook?", UNITN n°48
Walras L., "Éléments d'économie politique pure ou théorie de la richesse sociale", Economica, Paris
(1988)
Zahavi Y., "Travel Time Budget and Mobility in Urban Areas", Final Report 1974, Department Of
Transportation (1974)
8. APPENDIX - US DATASET

25. Daniel SALLIER 25
The following table includes all the data which has been used for modelling the US domestic air
market.
Domestic Air Traffic data and revenue, result from the concatenation of 3 different sources (Airline
Monitor for 1974-1995 period, BTS for 1990-2006 period and BTS for 1996-2010). The issue is
that while Airlines Monitor and 1990-2006 BTS data were overlapping "nicely", 1990-2006 and
1996-2010 BTS time series show a difference of about 10 to 15 M pax/year (2% data discrepancy)
between the 2 times series for the same years. In order to homogenise traffic data up to 2009, we
decided to add the 1996-2010 time series absolute annual pax increment to the 2006 annual data of
the 1990-2006 value.
Those data can be found on:
Bureau of Economic Analysis (BEA), National Economic Account, Gross Domestic Product
(GDP), internet site: http://www.bea.gov/national/index.htm#gdp
Bureau of Labor Statistics, Consumer Price Index, CPI databases, internet site:
http://www.bls.gov/cpi/data.htm
Bureau of Transport Statistics (BTS), "U.S. Air Carrier Traffic Statistics Through February 2010",
internet site: http://www.bts.gov/xml/air_traffic/src/index.xml#CustomizeTable