E u r o E c o n o m i c a [611527]

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19

Tools for the selection of microeconomic from socioeconomic rentability

Mohammed EL KHOMSSI1, Ghizlane CHAIBI2

1University Sidi Mohamed Ben Abdellah, [anonimizat]
2 University Sidi Mohamed Ben Abdellah, [anonimizat]

Abstract . The aim in this research paper is to propose useful and practi cal criteria to answer the question of
targeting aid of a micro -economic structure: Generating Activity of Revenues type. To achieve this goal, an
attempt is made to construct macro and micro indices linking the capital of a Generating Activity of Revenue s
with the update, which allowed us to obtain:
– A socioeconomic partition of structures constituting the fabric of Generating Activity of Revenues.
– Indices based on the global rentability permitting to order these structures of this basis notion.
Keywords: generating a ctivities of revenues, macrostructure index, microstructure index, r entability, current
value.

1 Introduction and Objectives

According to the United Nations Programme for Development (UNDP), the proportion of people
living below the threshold of extreme poverty in the world has decreased from 43% in 1990 to 22% in
2008. Similar results to those of the World Bank report, which shows that more than 40 developing
countri es have recorded increases in the value of their Human Development Index (HDI) much higher
than those considered; which illustrates the reduction of poverty, and that the world has already
reached the main target of eradicating poverty prescribed by Millen nium objectives for the
development. However, 22.4% of the world's population lives on less than 1.25 dollar per day
whereas early 1980s, 52.2% lived under this threshold, i.e. 1.3 billion people always live under the
extreme poverty line, nearly quart er of the world's population (Banque, 2000/2001) (Kim, 2 avril
2013) (PNUD, 14 mars 2013) .
The reduction of poverty is becoming the priority orientation of public pol icy. From this perspective,
several strategies and programs to combat this phenomenon are implemented to revive economic
growth, to improve living conditions, to strengthen human capital, and to ensure sustainable
development1 (Lachaud 1, 2000) (Lachaud 2, 2000) .
This latter approach has lifted the simple optical balance in the short term which does appease the
problems and not solves them radically towards a broader vision of a long -term deve lopment. In this
framework, and to achieve sustainable social development, the Generating Projects of Revenues
represent a means in this way as they have multidimensional implications (Fikri, 22 octobre
2011) (economic, social, health, cultural …) on the population, weak economically and therefore
socially. In Assuming that we have financial resources (either by the state or by non -governmental

1 INDH is a social and economic policy program, as an example applied to Morocco.

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20 organization or private sector) a major problem lies on which can by translated in the following
questions:
– According to what criteria can we help a micro -project to help the poor in a sustainable
manner? (Majda Fikri, 2011)
– Can we achieve economic and social rentability at the same time?
In this research paper we have constructed several mathematical tools to contribute and answer these
two questions.

2 Scheduling criteria
2.1 Targeting help problem
One of the problems is that; for a decision maker to choose2 to finance fully or partially a GAR3 class
and justify his choice economically, we consider mathematical tools based on performance, rentability
and the impact on the population. An active member of GAR is an effective way to give a meaning to
the notion of sustainable development because th e main objective is to save the members of the GAR
of poverty, helping them develop rentable projects economically and hence choose among the projects
those that have the possibility to achieve a gain and get them integrated into a society of
production/co nsumption instead of help/consumption.
The latter situation becomes heavy and unbearable by public and private finances taking into account
the current world conjecture (Sen, 1993) (Sen, 2000) .

2.2 Criteria of ordonnement for GAR class from the current value
2.2.1 Current value obtained from the balance
The decision to help finance any legal microeconomic structures4 is based on its economic interest,
and therefore the calculation of rentability. This latter depends on the project and the involved costs
and the gains that provides. If the sum of gains is greater to the various costs, particularly of the
investment, it is natural to say that it is rentable.
Our goal in this section is t o build a tool to measure the profitability that allows to select from the
GAR possible, the one that characterizes this notion better.
Let C the capital invested by a structure of financial aid GAR, consider the investment generates cash
inflows rated R and outflows recorded D over the years with T is the estimated life
equip.
(1)
If we assume that is the value of the equipment end of period T and k the discount rate future
value , and if we denote by flow back to the year i and to the output stream, we get:

2 We consider a decision maker any moral or natural person who has the legal status to finance this type of
micro -economic structure.
3 In this research paper we refer to a generating activity of revenues by GAR.
4 There are several legal frameworks and laws to define a social GAR. The model example in Morocco is the
social cooperative or association whose objectives is non – profit. ..

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21 ∑

∑

so we can write (1) as:
∑

∑

This allows to deduce that the net present value of a investment is the difference between the net
gains updated and recorded net costs knowing , the economic balance gives:

2.2.2 Ordonnement criterion and construction of a canonical partition
Let set of GAR possible or alternatives and the net present value of the GAR j noted if the
decision maker chooses the viable GAR in the set he is choosing GAR which achieved:

This choice is possible because this value is the upper bound of a finite set . In an iterative manner
we set and } and choose the GAR which achieved:

We will continue the same way until the last of the GAR that: { }
knowing that and { } with and is reached
.
So we built the first criterion to order it allows you to choose the one which carries and
classify GAR's choice (4) this operation allows you to switch of to and order the whole
structure.
Consequently, we have the following partition:
⋃

with a sequence of sets decreasing satisfying:
and { }
with { } is the singleton sets by:

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22 { } ⁄

consequently the set of all GAR is written in this form
⋃{ }

this writing is a canonical ordered partition of .

2.3 Microstructure Criteria
It is possible to make a choice to support and invest in financing the all GAR of set , for taking this
one reference value 1 in the following manner:
Recall that C the global capital, and capital GAR, let

Definition [Profitability index of GAR ] is called index profitability micro -structure the ratio of the
current value future cash flows and capital:

Obviously this amount is set to the reference value of 1, indeed, changes sign and we have three
possible cases:
 It is an economic loss, but the GAR can achieve social gain (integration
population …).
 There is a loss of present value of the estimated capital or the costs of not
location and the cost of inflation, but this situa tion better than the first.
 GAR realize an economic gain.
We summarize the relationship between the sign of and the notion of economic gain in the
following table:
Economic Gain
negative not realizable
null neutral situation
positive realizable

For social gain, all situations are open in three cases. Subsequently, our goal is to exploit index
profitability to build another partition structure

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23 2.4 Algorithm optimal or admissible choice
A natural question arises, how to define an order of priority between GAR finance performing
all ?
From the previous entries, we have a partition as follows:
⋃̇ ⋃̇
which:
{ }

The profitability index is used to construct a partition and distinguish that which is rentable in
economic meaning, that is to say .
Hence, with this choice we have built a second criterion of partition of , based on the calculation of
the profitability index .
It is obvious that if we apply the first criterion and the second, we obtain a highly interesting
classification between GAR achieving a better discount with index profitability than 1, that is to say, if
we couple the two criteria, we get the GAR satisfying the two following conditions
simultaneously:

The first criterion helps us to choose the GAR verifying the best value discount, the second helps us to
choose GAR perfor ming the best index of profitability, this way we get a finer classification. Also we
note that:

that:

and
It means that the selection of GAR by the first criterion is the structure . Therefore, if there is ,
which is an optimal choice, we obtain:

but obviously, it is possible that does not exist in this if we work in the class and the
following choice of decision maker (permissible range), we search the GAR verifying:

or

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24 We can search the GAR which verifies the best value discount or GAR verifying the best index
profitability using a programming language, if we use the Matlab example, the 'sort' to sort the
elements of a vector as follows:
Algorithm for selecting the opti mal or admissible GAR
>> V = [ ] the capture of current
>> I = [ ] input indices profitability
>> [OV, PV] = sort (V)
>> [OI, PI] = sort (I)
with: – OV: contains the sorted vector V
– OI: contains the sorted vector I.
– PV: positions contains elements of the vector OV in the vector V.
– PI: positions contains elements of the vector OI in the vector I.
if OV(1) = OI(1) then = OV(1) and we have a solution optimal, if not the GAR OV (1) realized the
best value discount and GAR OI(1) realized the best index profitability.

2.5 Macro -structure Criteria
The above analysis is a comprehensive analysis of funding, which can or c an't be profitable
economically , this stage depends on the writing index j, our goal now is to construct a m acro-structure
index that is to say that which characterizes 5 In fact we can build on the an index of all
structure from which the following definition:
Definition [Profitability index of a class GAR] Profitability index is called macro -structure of a class
of GAR index defined as follows:
∑

∑

Remark: One of the advantages of this index is a to compare overall different economic structures, in
fact, to compare the profitability of the structure with the structure of we consider the
difference:

the sign of this quantity clarifies that which is profitable overall. otherwise said, this index can allow is
to say that in a region GAR type Agronomy better than Craft or vice versa. For example, table 2 is an
extract from the Statistical Yearbook of the region of Al Haouz Marrakech (Royaume du Maroc,
2010) gives the social capital of cooperatives of wood and plants as well as the export volumes of
2010 :
Craft Capital (DH) Export (DH)
Wood 2736257 27863962

5 The rationale of the test macrostructure because it characterizes , without addressing its elements one by one. It is a way
to compare the profitability GAR with another structure (an GAR agricultural structure and other structure GAR crafts for
example).

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25 Plants 1084769 5709996
Table 2
recall that therefore ∑

∑

If we assure that the statistics of GAR are identical to those of cooperation in macroeconomic , and if
we assure that the gains are reduced only to export, we obtain the following table :
Craft
Wood 10.18
Plants 5.26

We note that for the region of Marrakech -Tensift -Al Haouz it is better to encourage and sponsor
wood GAR.
Proposition: For each structure GAR , there are two critical values and as:

This result is simple, since it suffices to take:

and

with:
and

2.6 Classification of the elements of the structure
Our aim in this section is to classify all of generating activities of revenues constituting the class
based on their net present values to achieve this goal, we introduce two following definitions:

Definition [absolute gap] For every we define the value as absolute difference of profitability
by:

is positive by construction, this gap provides information on as follows:
If tends to 0, we deduce that the net present value of the GAR is almost constant for all elements of
whereas, if is very large compared to 0 ( , it reflects that there are some GAR in more
profitable than others, an unidentified interest of this gap.
In the economic sense, that tends to 0 or is very large compared to 0 is a little vague, for greater
accuracy, we introduce the relative g ap defined by:
Definition [Relative gap ] For each , we define the value as relative gap of profitability by:

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26 Obviously, from the definition of we have

Classification GAR classes using the relative gap
For a class GAR , if is nearest to 1, then there is a strong variation between different GAR, that is
to say, there are those which more profitable than others whereas, if tends to 0, we conclude that the
net present value is almost constant for all GAR for structure
If we have finite classes GAR , let ⋃
, and let relative gap on each by
we have

practically, we can introduce parameter with to decompose the range [0, 1] as:

consequently we group in subclasses as follows:

with
⋃ ⋃ ⋃
Using this combination, we have classified all GAR of class using the notion of relative difference
in under sets, and we can say that:
 GAR of almost the same net present value.
 GAR of have a significant variation in the present net value.
 For GAR the expert can estimate their variations.

Generalizatio n Subdivision:
For more accuracy, the expert may consider further subdivision of [0, 1], it can work with a regular
subdivision in this case we have:

⋃

as it may take a non -regular subdivision, that is to say:
⋃

with and for all j=2,.., M -1 and
In this case, we have:

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27 ⋃

with
⋃

 GAR of almost the same present net value.
 GAR of have a significant variation in the present net value.
 For all GAR of with 2< j< M -1the expert can estimate their variations.

3 Construction of a global index

3.1 Concept of a global index
According to the definition the profitability index , projects with are projects that provide
only the same initial value capital, that is to say, the value of cash flows discounted. Therefore on the
threshold value of 1 is between the projects to finance and those that are rejected, if we consider only
the economic objec tive, knowing that owned ̇ are GAR that can check a social gain,
bat not an economic one.
The dispersion around 1 is an important factor to analyze information on the nature of the GAR. If we
inspire the statistical techniques (Chabert, 1989) (SILLARD, 2001) a transcript of the idea of the error
of the least squares sense, helps us define a global index as follows:
Definition: Called global index of a structure GAR, index defined by:
∑

This amount is used to evaluate the variance of indices of profitability of all GAR projects in relation
to the threshold 1, it measures the dispersions and the accumulation around 1. Where the value of this
index in the following sense: It is independent of the GAR, more; it measures the difference between
the average value, so as to give information on the decision to couple a social gain to an economic
one.
From the equation (5) also written as form:
∑

recall that

and

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and put

̅̅̅

̿

Lemma: Let indicates an overall structure of GAR
– If and have the same sign then:
̅̅̅ ̿
– If and have different signs then:

̅̅̅
Proof:
According to the signs of and we have the following inequalities:
– If then

– If then

– If and have different signs then:
((
)
(
)
)
3.2 Dominant class of GAR
The objective of this section is to provide a technique to select the dominant class of GAR, for this,
recall that is written as follows:
̇ ̇
which allows to obtain:

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29 ∑
∑
∑

The class has weight in this writing thus:
∑
∑

put defined by the ratio:

we have:

If is close to1 we can say that the class is dominant, in the following sense:

and

Or is equivalent to saying that for all of , tends to 1, which gives:

in a similar way we can analyze the case where is nearest to 0. According to the value of the latter,
we have:
̇ ̇
with class GAR dominant in the sense described before, and class GAR characterized that

Consequently, we have a quantized value to know the dominant subclass of class GAR, more, from
the two conditions:

and

we get a way to make a strategic decision select the best GAR.

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30 4 Conclusion and perspectives

The research constructed micro -structure and macro criteria based on the index of profitability to give
a quantified argument for the decision maker of relie on pragmatic and optimal decisions to help set
of generating activities of revenues, according to the strategy he chooses, not subjectively, and not
politically .
A numerical simulation is used to select, in all indices proposed an optimal choice or admissible
activities generating revenue that the decision maker search to finance. In the prospect o f obtaining
reliable and practical data from ministries6, we propose, in a future work, a numerical simulation with
interpretations.

5 References

Banque, M. (2000/2001). Rapport sur le développement : Combattre la pauvreté. Paris: Éditions Eska, 2001, XIV -38l p .
Chabert, J. L. (1989). Gauss et la méthode des moindres carrés" In: Revue d'histoire des sciences.
Fikri, M. (22 octobre 2011). Thèse du grade Docteur: Planification des ressources.
Kim, J. Y. ( 2013, Avril 2). Discours du président du Groupe de la Banque Mondiale. l'université de Georgetown .
Kim, J. Y. (2 avril 2013). l'université de Georgetown.
Lachaud 1, J. -P. (2000). Les déterminants de la réduction de l'inégalité et de la pauvret é en Mauritanie : une approche
micro -économique. Groupe d'Economie du Développement de l'Université Montesquieu Bordeaux IV.
Lachaud 2, J. -P. (2000). Echelles d'équivalence et différentiel spatial de pauvreté et d'inégalité au Burkina Faso. Groupe
d'Econom ie du Développement de l'Université Montesquieu Bordeaux IV.
Majda Fikri, M. E. (2011). Proposal of a Semi Fuzzy Poverty Index. EuroEconomica , .
PNUD. (14 mars 2013). Rapport sur le développement humain 2013 : L'essor du Sud. New York.
Royaume du Maroc, H. c. (2010). Annaire statistique de la région de Marrakech Tensift Al Haouz.
Sen, A. (1993). Capability and Well -Being . The Quality of Life. pp. 30 -53. .
Sen, A. (2000). l'exclusion sociale: Concept, l'application et examen. Banque asiatique de développem ent.
SILLARD, P. (2001). Estimation par moindres carrés.

6 Ministry of Interior, Ministry of Solidarity, Women and Social Development, Social Division Provinces,
micro -credit Groups …