*Periodicity.:*

**October - December 2014***e-ISSN......:*

**2236-269X** **ESTIMATION OF THE INFORMAL ECONOMY: AN
APPLICATION OF FUZZY SETS**

*Vinícius Ximenes Provenzano*

*IBMEC, Brazil*

*E-mail: viniciusxp@gmail.com*

*Walter Gassenferth*

*IBMEC, Brazil*

*E-mail: walterg@quantiac.com*

*Maria Augusta Soares Machado*

*IBMEC, Brazil*

*E-mail: mmachado@ibmecrj.br*

*Submission: 22/04/2014*

*Revision: 10/05/2014*

*Accept: 24/05/2014*

ABSTRACT

Data on “Informal Economy”
magnitudes are relevant for the establishment of adequate micro and
macroeconomics but, there are very few indicators in this respect. The goal of
this paper is to generate a temporal series that would describe the informal
economy in

## Keywords: informal economy, fuzzy logic, estimation

**1. ****INTRODUCTION**

Data on the size of the global economy of any country or
bloc, and their

economic indicators are the result of the study and calculation of statistics,
on trade, production and investment of the object of study. But there is one
part of the economy that is not captured by economic indicators and has long
been considered of little relevance to the formation of the economy in toto, this
part is commonly called the Informal Economy.

But there is much discussion about the accuracy of
economic indicators of this phenomenon. The very definition of the informal
economy is also discussed, which makes it more difficult to measure it. The
importance of having a measure of the informal economy is also measure the
effectiveness of measures taken to fight it and not stimulate their growth,
thus generating an increase in government revenue and benefits to society as a
whole.

The theory of fuzzy sets, an extension of the classical
theory of sets, has been shown effective in many different fields with data
using subjective criteria. Using this tool for the economy is still little
explored, and the need to find a good measure for the informal economy appears
as a fertile field for experimentation. Draeseke and Giles (2002) made the case
for New Zealand.

The objective of this paper is to generate an annual
series that describes the informal economy in Brazil for the period studied.

Fuzzy sets were used as a theory tool. It is part of the
informal economy that all economic activity that is outside or legality of the
tax. Fraudulent activities, smuggling and drug trafficking are part of the
informal economy as much as the omission of income, rent irregular rent or exchange of goods and
services.

**2. ****FUZZY SETS**

The theory of Fuzzy Sets was developed by electronics
engineer Zadeh (1975), professor at the University of California. In the 70's
began to be widely adopted mainly in the areas of data classification, expert
systems, decision analysis, robotics, pattern recognition and time series
forecasting.

For example, the phrase "high unemployment" can
be interpreted in many different ways depending on the situation. In Sweden,
where the unemployment rate used to be 3%, can be interpreted as a high
unemployment rate any higher than 5%. In other countries, however, the
unemployment rate cannot be considered high unless it exceeds 10%. In terms of
fuzzy logic, "high unemployment", as it is called a subjective category.

In contrast to conventional probability theory, the
theory of fuzzy sets

uses language to describe the uncertainty in the real world. Thus allows the
use of linguistic expressions such as "

The degrees of membership values are intermediate between
the values of true (0) and fake (1) in traditional Boolean logic. In fact, the
Boolean logic is a particular case of fuzzy logic, the case in which the
degrees of relevance are the extremes. The degrees of membership are assigned
through a membership function.

Fuzzy Reasoning or approximate reasoning is an inference
procedure that derives conclusions of a set of fuzzy if-then rules of known
facts.

The knowledge of the phenomenon is expressed through
statements like: "if (a set condition is satisfied) then (we can infer a
set of consequences)" (OLIVEIRA et al. 2007).

**3. ****METHODOLOGY**

The use of fuzzy set theory as a tool in economics is
still very small. Draeseke and Giles (2002) propose its use to measure the
degree of informality in the economy. The hypothesis that motivates this work
is the ability to measure the informal economy, an unobservable variable of the
economy through two entrances. The method of approach is deductive and it is a
case study in Brazil.

The proposed technique is based on Mamdani Fuzzy
Inference System.

This paper aims to produce an experimental quantitative
research as having premised on the existence of a causal relationship between
variables. The samples were obtained from official data released by the IBGE
and the frequency of time series is annual.

The first observation of each series is about the year
1974 and the last to 2002, totaling a period of 28 years.

Draeseke and Giles (2002) had used as input variables in
their fuzzy inference system the degree of regulation of the economy and taxes.
In

The first step was to obtain the following time series
between 1974 and 2002:

- Exports
- imports
- GDP
- Total Gross Tax (% GDP)

From the first three series it was generated the index of
the degree of openness of the first input variable. The total tax is the second
input variable.

The following charts present the series that were used to
obtain the first input variable.

Figure
1: GNP (a), Imports (b) , Exports (c) ,
d) Total Gross Tax (% GDP) :1974-2002

Next graph presents tax load rate in relation to GNP
(Grow National Product).

Figure
2: tax load rate in relation to GNP (Grow National Product)

Each of the input variables was modeled with five
membership functions, according to its degree of magnitude.

Thus both the variable opening of economy as variable
Tax were modeled with the following
membership functions: . Very Low, Low, Normal, High and Very High.

Trials were made on the modeling of input variables using
triangular , Gaussian and Sigmoid membership functions.

The rule base used in this paper was that proposed by Draeseke
and Giles (2002).

Removing the weights proposed by Draeseke and Giles, one
can obtain a new rule base. The option of having two sets of rules reinforces
the experimental feature of this paper, to compare the results obtained using
two sets of rules. The first basic rule is called Heavy Rule Base and the
second Not Heavy Rule Base. Then it is created a new table containing the new
rule base, where the only change from the previous table is that all rules have
the same weight of 1. As the input variables, the output variable, informal
economy is modeled with five membership functions also called: Very Low, Low,
Medium, High, Very High. It were used Gaussian membership functions to model
the output variable.

The rules are shown in table 1.

Table 1- rule base used in this paper

Openness of economy |
Tax burden |
Level of informality |
Weigh |

Very high |
Very high |
Very high |
1 |

Very high |
High |
Very high |
0.8 |

Very high |
Normal |
Low |
1 |

Very high |
Low |
Low |
0.8 |

Very high |
Very Low |
Medium |
0.8 |

High |
Very high |
Very high |
1 |

High |
High |
High |
1 |

High |
Normal |
High |
0.8 |

High |
Low |
Medium |
1 |

High |
Very Low |
Medium |
1 |

Medium |
Very high |
High |
1 |

Medium |
High |
High |
0.8 |

Medium |
Normal |
Medium |
1 |

Medium |
Low |
Low |
0.8 |

Medium |
Very Low |
Low |
1 |

Low |
Very high |
High |
1 |

Low |
High |
Medium |
1 |

Low |
Normal |
Low |
0.8 |

Low |
Low |
Low |
1 |

Low |
Very Low |
Very Low |
1 |

Very Low |
Very high |
Medium |
0.8 |

Very Low |
High |
Low |
0.8 |

Very Low |
Normal |
Low |
1 |

Very Low |
Low |
Very Low |
0.8 |

Very Low |
Very Low |
Very Low |
1 |

**4. ****RESULTS AND CONCLUSIONS**

Analyzing the graphs modeled using triangular and
Gaussian membership functions it can be seen that there is not a difference
between the two series (using unitary and different weights).

It can be seen in Figures 3 and 4 that when using
triangular membership functions the informal economy decreases from 4 (1974) to
1 (1980) and grows a little bit in 2000th but dacays after, it can be said that
it is stable around 5 during all 28 years and when using Gaussian membership
functions the informal economy is a constant with value 15 from 1974 to 2004.

Figure
3: Informal Economy modeled using triangular membership functions

Figure
4: Informal Economy modeled using Gaussian membership functions

For other kind of membership functions (sigmoid and
trapezoid) the values are near to the Gaussian.

It has been done non parametric statistics tests to
verify if there exists differences between these obtained values and the
conclusion was that there exist statistical differences with 95% of confidence.

This work may be used as a starting point for improving
the method of

Measurement of the Informal Economy. The addition of new input variables, for
example, the unemployment rate and an extension of the rule base for the
inclusion of new variables can show relevant results. Another search is being
done using an econometric study for the most relevant variables and then
rewrite a new rules base to compare with these results.

According to the Brazilian Institute of Geography and
Statistics (IBGE), Brazil were more than 10 million urban informal enterprises
in 2003 , and moved over 17 billion dollars. The number corresponds to more
than half of all microenterprises in Brazil.

This kind of economy is directly linked to the Gross
Domestic Product of the country, because many goods are sold and obtained
without taxes payments. To get an idea if everything was formalized in Brazil, our
GDP would be 30% greater than it is today.

According to Holanda (2012), between 2006 and 2011,
informal economy has fallen from 20.2% to 17 % " in large part , this
decline is explained by the significant increase in the formal labor market in
recent years , a consequence of the good performance of the Brazilian economy
in the period , even during the 2009 crisis ".

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