Research ArticleA Study of the Impact of Underground Economy on Integral TaxBurden in the Proportional Growth Model under Uncertainty

Akif Musayev 12 and AygunMusayeva3

1 Institute of Economics Azerbaijan National Academy of Sciences Baku Azerbaijan2Near East University Nicosia Northern Cyprus Mersin 10 Turkey3Azerbaijan University Baku Azerbaijan

Correspondence should be addressed to Akif Musayev akifmusayevgmailcom

Received 3 August 2017 Accepted 15 February 2018 Published 1 April 2018

Academic Editor Rafik Aliev

Copyright copy 2018 Akif Musayev and Aygun Musayeva This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Economic processes are naturally characterized by imprecise and uncertain relevant information One of the main reasons isexistence of an underground economy However in existing works real-world imprecision and uncertainty of economic conditionsare not taken into account In this paper we consider a problem of calculation of a taxation base to assess tax burden forproportionally growing economy under uncertainty In order to account for imprecision and uncertainty of economic processes weuse the theory of fuzzy sets A fuzzy integral equation is used to identify an integral tax burden taking into account the contributionof the underground economy for a certain financial (tax) year It is also assumed that dynamics of gross domestic product aremodeled by fuzzy linear differential equation An optimal value of tax burden is determined as a solution to the considered fuzzyintegral equation An example is provided to illustrate validity of the proposed study

1 Introduction

The current stage of economic development is characterizedby the influence of several fundamental factors In scientificliterature these factors are divided into clusters according totheir nature stages coverage and so on

According to Balatsky [1] these factors split into threegroups generality uncertainty and economic impact It hasbeen noted that they have formed new producing capacity byinfluencing the structure of the economic system the normsand trends in the economy

One of the important factors of economic development isthe tax burden of the economy Economic entities transmit aportion of their income to state or local self-governing bodiesin the form of taxesThis transfer of financial resources limitsthe ability of consumers to choose between products andability of producers to choose between production factorsthus creating additional burden on economic entities

Determining the tax burden in line with the realityformed by the economic system is one of the most difficultproblems It is connected with the fact that the tax system

should cover the public needs andon the other hand stimulatethe interests of taxpayers in entrepreneurial activity

In this sense the tax burden is considered as the qualityof the countryrsquos tax system and the overall business environ-ment

The study of the impact of the tax burden on economicactivity and the statersquos financial strength remains the mostpressing issue of economics and economic policy In ages ofglobalization this problem attracts attention of economicsexperts as well as institutions of different countries

Public administration and legislation system also have asignificant impact on economic growth due to share of overallconsumption in gross domestic product (GDP) aggregateinvestment public spending and net exports For examplethe introduction of a new taxation regime to fixed capitalinvestment and repatriation of income in tax legislation canaffect transfer of new technologies research and develop-ment works as well as import-export processes

It is well known that the tax burden of economy isexpressed in terms of separate taxes established by tax

HindawiAdvances in Fuzzy SystemsVolume 2018 Article ID 6309787 5 pageshttpsdoiorg10115520186309787

2 Advances in Fuzzy Systems

legislation and at the same time as an integral quantitativeexpression of the cumulative effects of these taxes (integraltax)

In modern economic literature depending on the pur-pose of the research the tax burden is differentiated bythe maximum average actual effective consumption publicexpenditure and the increase in tax rates There are variousapproaches to calculating the tax burden In practice themost commonly used approach by the public tax policyinstitutions is to set the tax burden as a ratio of aggregatedtax revenues to GDP

Generally these approaches are described in variousscientific bibliographies For example in [2] the tax ratesare described as marginal rate final tax rate and economic(effective) tax rate

At the microlevel the tax burden of an economic entity isdetermined as the rate of paid taxes to the total income gainsarising from product produced and the service rendered It isalso possible that the tax burden is determined as the ratio ofthe amount of taxes paid to the tax base

The literature review allows coming to conclusion that allabove-mentioned approaches are far from a universal one Inthis regard identification of this particular economic factorremains an important issue of modern economic research

Looking at the economic growth theories in the historicalcontext the first remembering theory is of mercantilism(15thndash17th centuries) The theory of mercantilism shows theaccumulation of wealth (gold and other precious metals) asthe main source of economic growth [3] In the followingperiod neoclassical economists criticized the New Keyne-sians argued that the cause of economic growth was not justin the frame of unused production factors but also techno-logical advances and increased productivity through inno-vations in production management Neoclassical economistsbelieved that economic growth would be achieved in a freemarket economy The theory proposed by the Nobel Prizewinner Robert Solow based on the Cobb-Douglas produc-tion function also linked three main sources of economicgrowth investments workforce and technological progress[4] Along with this model saving rate plays a key role informing capital reserves and increasing production

In the 1980ndash1990s significant progress was observed ineconomic growth theories This trend was announced as aldquonew economic theoryrdquo Economists have examined someweaknesses of the neoclassical economists in particular theSolow model and made proposals to increase the impact ofthe government to long-term economic growth According tothe Solowmodel the impact of the government on economicgrowth is considered as very insignificant However adherersof the endogenous economic growth theory suppose thatscientific and technological progress plays a crucial rolefor economic growth as endogenous factors Romer [5 6]and Lucas Jr [7] argue that technological innovations areprimarily due to technology development and human capitalinvestment Technological progress in endogenous growththeories is considered as the only viable cause of economicgrowth in the long-term perspective Similar reasons wereraised in theories of Grossman and Helpman [8] and Aghionand Howitt [9]

In [10] the tax burden and economic development in caseof the EUcountries are discussedThere are differentmethodsfor evaluation tax burden and economic development Theseinclude such research methods as systematic logical andcomparative analysis Statistical methods are also used inincluding descriptive statistics hierarchical cluster analysisand correlation analysis Empirical analysis is based on thedata of the EU countries It is shown that the tax burden oncapital and consumption is higher in the very high economicdevelopment countries but implicit tax rate on capital ishigher in the case of countries with lower GDP growth

It is known that the excess burden of taxation showsefficiency cost associated with taxation In [11] excess burdenis measured by the area of the associated Harberger triangle[12] Excess burden task is discussed in Arnold Harbergerrsquospioneering work in the 1960s which measured the costs oftax distortions to labor supply saving and other economicdecisions [13 14]

Using a simple input-output model in [15] the authorestimates the revenue impact US tax burden state-by-stateimpact of this proposal on tax burdens employment andeconomic output Also authors show that elimination of thededuction for the oil and gas industry would result in highercorporate tax burdens for domestic oil and gas companiescompared to other US manufacturing industries In realityUS oil and gas companies could not hope to keep pacewith competitors in countries with lower tax rates Moreoverincreasing corporate taxes on the US energy sector increasethe costs of production and may reduce the resources

Determination of the optimal size of tax burden is veryimportant and difficult scientific and practical problem Themain goal of [16] is to determine the optimal of tax burdenfor the economy of Georgia Optimally determined taxes canplay very important role for sustainable development of eco-nomics For this reason author used qualitative quantitativecorrelationregression analysis Tax burden for the economyofGeorgia depends on size and type of activities of enterprisesand should range from 136 to 176

The influence of optimal tax burden on economic activityand production capacity is discussed in [17] In this paper thetax policy and an optimal tax burden are described

Let usmention that information related to economic pro-cesses is naturally characterized by uncertainty and impre-cision due to a series of reasons one of which is existenceof an underground economy In this paper we consider aproblem of calculation of a taxation base to assess tax burdenfor proportionally growing economy under uncertainty Inorder to account for imprecision and uncertainty of economicprocesses we use the theory of fuzzy sets A fuzzy integralequation is used to identify an integral tax burden taking intoaccount the impact of the underground economy for a certainfinancial (tax) year An example is provided to prove validityof the proposed study

The paper is organized as follows In Section 2 we providea preliminary material to be used in the study includinga definition of a fuzzy number-valued function integral ofa fuzzy number-valued function and other concepts InSection 3we describe the proposed fuzzy approach to identifyan integral tax burden taking into account the contribution

Advances in Fuzzy Systems 3

of the underground economy A method to assess impact ofunderground economy is described in Section 4 An exampleillustrating the proposed fuzzy approach is considered inSection 5 Section 6 concludes

2 Preliminaries

Denote 1198641 is the set of fuzzy numbers defined over the realline R

Definition 1 (a fuzzy number-valued function [18 19]) Amapping 119891 119879 rarr 1198641 is referred to as a fuzzy number-valuedfunction of a real variable

Definition 2 (see [18 19]) A function 119891 119879 rarr 1198641 isreferred to as integrably bounded if there exists such real-valued function ℎ that 119909 le ℎ(119905) forall119909 isin supp119891(119905)

A definition of measurability of a fuzzy number-valuedfunction of a real variable is given in [18] In [18] it isshown that strongly measurable and integrably boundedfuzzy number-valued function is integrable

Definition 3 (an integral of a fuzzy number-valued function[18]) Let 119891 119879 rarr 1198641 An integral of 119891 on 119879 = [119886 119887]denoted as int

119879119891(119905)119889119905 or int119887

119886119891(119905)119889119905 is defined in terms of 120572-cuts

as follows

[int119879119891 (119905) 119889119905]120572 = int

119879119891120572 (119905) 119889119905 = int

119879120593 (119905) 119889119905 120593 119879

997888rarr 119877119899 minusmeasurable choice for 119891120572 forall120572 isin (0 1] (1)

Definition 4 (strongly generalized differentiability [20]) Let119891 (119886 119887) rarr 1198641 and 1199050 isin (119886 119887) We say that 119891 is stronglygeneralized differentiable at 1199050 if there exists an element1198911015840(1199050) isin 1198641 such that

(i) for all Δ119905 gt 0 sufficiently small exist119891(1199050 + Δ119905) minusℎ 119891(1199050)119891(1199050) minusℎ 119891(1199050minusΔ119905) (ie the length of diam((119891(119905))120572) increases)and the limits (in the supremummetric [18 19])

limΔ119905rarr0+

119891 (1199050 + Δ119905) minusℎ119891 (1199050)Δ119905 = lim

Δ119905rarr0+

119891 (1199050) minusℎ119891 (1199050 minus Δ119905)Δ119905

= 1198911015840 (1199050)(2)

or(ii) for all Δ119905 gt 0 sufficiently small exist119891(1199050) minusℎ 119891(1199050 +Δ119905) 119891(1199050 minus Δ119905) minusℎ 119891(1199050) (ie the length of diam((119891(119905))120572)

decreases) and the limits (in the supremummetric [18 19])

limΔ119905rarr0+

119891 (1199050) minusℎ119891 (1199050 + Δ119905)(minusΔ119905) = lim

Δ119905rarr0+

119891 (1199050 minus Δ119905) minusℎ119891 (1199050)(minusΔ119905)

= 1198911015840 (1199050) (3)

minusℎ denotes Hukuhara difference

Definition 5 (possibility measure [21 22]) Given two fuzzysets defined in the same universe of discourse 119883 a funda-mental question arises as to their similarity or proximityThere are several well-documented approaches covered in theliterature One of them concerns a possibility measure Thepossibility measure denoted by Poss(119860119883) describes a levelof overlap between two fuzzy sets and is expressed in the form

Poss (119860119883) = sup119909isin119883

[119860 (119909) and 119883 (119909)] (4)

where and is a 119905-norm Computationally we note that thepossibility measure is concerned with the determinationof the intersection between 119860 and 119883 that is followed bythe optimistic assessment of this intersection It is done bypicking up the highest values among the intersection gradesof 119860 and119883 that are taken over all elements of the universe ofdiscourse1198833 Fuzzy Model of a Tax Burden

First let us provide an explanation of important economicindicators considered in the paper

As a rule a tax burden 119906 is defined as the following ratio

119906 = 119879120585119870 (5)

where 119879 is tax revenue and 120585119870 is equity capital load takinginto account taxes

A tax burden leads to an economic system transition from119910(119905) state to 1199101(119905) state Assume that at the beginning of aconsidered year GDP depends on a value of a fixed capital

119870119894119901(119905) = 119875(119905minus1) (119870119894119901(119905minus1) minus 119877119894119901(119905minus1) minus 119860 (119905minus1)) (6)

where 119877119894119901 is investment 119860 = 120579119870 depreciation-amortizationof119870 value loaded share of core capital 120579 is amortization rate119875 is deflator

It is obvious that information related to some parametersof underground economy and economic growth in general isimpreciseThus it is more adequately to use fuzzy models fordescribing tax burden in such conditions

A tax burden 119906(119905) leads to an economic system transitionfrom fuzzy state 119910(119905) to fuzzy state 1199101(119905)

1199101 (119905) = 119906 (119905) 119910 (119905) (7)

Let us use the following terms and symbols 119905 is financial(taxation) year 119910(119905) is the income of the economic systemfrom the production and service rendered by the business1199101(119905) is posttax income of an economic entity 119906(119905) is thetax burden of the economic entity in the financial (tax) year120585 is equity capital load taking into account the taxes isthe fuzzy value of the growth rate of the economy 120578 is thefuzzy value of the underground economy weight 120591 is periodfor tax liabilities The following fuzzy equation comes fromeconomic balance conditions

int1205910[119910 (119905) minus 1199101 (119905)] 119889119905 = (1 + 120578) 120585 (8)

4 Advances in Fuzzy Systems

or

int1205910[119910 (119905) minus 119906 (119905) 119910 (119905)] 119889119905 = (1 + 120578) 120585 (9)

Now we need to analyze the form of 119910(119905) As we considerproportional economic growth model under fuzzy environ-ment dynamics of 119910 can be adequately described by thefollowing fuzzy differential equation [18ndash20] under (ii) typeof strongly generalized differentiability (Definition 4) [20]

119889119910119889119905 = 119910 (10)

It is known that the solution of (10) is

119910 = 1199100119890119905 (11)

which can be described in 120572-cuts as follows1199101205721 = 119910012057211198901205821205721 1199051199101205722 = 119910012057221198901205821205722 119905

(12)

Thus given solution (11) we can transform (9) into thefollowing integral equation

int1205910(1 minus 119906 (119905)) 119890119905119889119905 = 120585 (1 + 120578) (13)

Given the obtained fuzzymodel of tax burden we need toconsider a problem of determination of 119906(119905) that solves (13)Wewill consider determination of 119906(119905) in the class of constantfunctions 119906(119905) = 119906 119906 = const As (13) is fuzzy the equality ofthe left and the right hand sides should be considered in termsof a degree As a measure of a degree of equality we will usethe possibility measure Then the considered problem can beformulated as follows

119875(int1205910(1 minus 119906 (119905)) 119890119905119889119905 120585 (1 + 120578)) 997888rarr max (14)

st 119906 (119905) = 119906119906 = const (15)

where119875 is the possibilitymeasure In otherwords it is neededto find such a value of 119906 = const for which the possibilitymeasure of equality of the left and the right hand sides ismaximal

4 Expertsrsquo Group Opinion Based Estimationof the Impact of Underground Economy onTax Burden

Experience has shown that comprehensive indicators of aneconomy system are not quite precise due to uncertainty ofmajor factors One of the most important of these factors isthe weight of the underground economy It is of particularimportance if one takes into account comprehensive burdenfor the economyNowonder assessing of this factor is in focusof major modern surveys

00102030405060708091

091 094 097 1 103 106 109 112088

Figure 1 The fuzzy values of the left hand side (dashed curve) andthe right hand sides (solid curve) of (14)

In [23] the theoretical approaches to the definition of thehidden economy taxation base are widely analyzed specifiedinto three groups and recommended to use the volume ofGDP as the tax base However there is no comment on howto determine the specific weight of the hidden economyObviously the problem of determining both the specificweight of hidden economy and tax base comprised its taxburden is one of the pending problems

In this study the load factor of the main capital deter-mining the loaded part of the fixed capital 120585 is specifiedby the quantity 120578 As an underground economy is of latentcharacter an adequate estimation of its impact 120578 can bebased on expertsrsquo opinions described in natural language andformalized by using fuzzy sets

The opinion of expert group on the impact of changes intax legislation on tax revenue is computed by using aweightedaverage based aggregation

120578 = sum119899119894=1 119902119894 sdot 119908119894sum119899119894=1 119908119894 119894 = 1 119899 (16)

where 119902119894 is a fuzzy evaluation of 120578 provided by 119894th expert and119908119894 is a fuzzy confidence degree of 119894th expert

5 An Example

Consider the following example Let and = (10 30 50)be a triangular fuzzy number (TFN) describing fuzzyvalue of the growth rate of hypothetic economy let 120578 =(032 034 035) be a TFN describing expert group-basedcomputed by using (16) and 120585 = 0737 120591 = 1 in(13) It can be shown that the left hand side of (13) is anintegrable fuzzy number-valued function of a real variableThus we can consider a problem (14) to determine 119906 Byusing numerical techniques we have found 119906 = 015 as thesolution of (14)which provides119875(int120591

0(1minus119906(119905))119890119905119889119905 120585(1+120578)) =

096 This value of possibility measure can be consideredas sufficiently suitable from practical point of view The leftpart of (13) approximated by TFN int120591

0(1 minus 119906(119905))119890119905119889119905 =

(089 099 11) and the right part described by TFN 120585(1+120578) =(09733 09857 09982) are shown in Figure 1

Advances in Fuzzy Systems 5

6 Conclusion

In this paper we use fuzzy integral equation to model theway tax burden induces transition of state of economy Anoptimal value of tax burden is considered as a solutionto fuzzy integral equation The use of fuzzy sets helpsto model imprecise information including expertsrsquo opinionbased evaluation of an underground economy weight Anexample is used to illustrate the proposed approach In theexample a tax burden 119906 = 15 is found as a solution tothe fuzzy integral equation with possibility degree 096 Theobtained result shows validity of the proposed study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y V Balatsky ldquoA mechanism of dependence between innova-tions and economic growthrdquo Science Innovations and Educa-tion vol 2 pp 182ndash198 2007 (Russian)

[2] I A Mayburov Theory of Taxation Advanced Course UnityMoscow Russia 2011

[3] J McDermott ldquoMercantilism and modern growthrdquo Journal ofEconomic Growth vol 4 no 1 pp 55ndash80 1999

[4] RM Solow ldquoA contribution to the theory of economic growthrdquoQuarterly Journal of Economics vol 70 pp 56ndash94 1956

[5] P M Romer ldquoIncreasing returns and long-run growthrdquo Journalof Political Economy vol 94 no 5 pp 1002ndash1037 1986

[6] P M Romer ldquoEndogenous technological changerdquo Journal ofPolitical Economy vol 98 no 5 pp S71ndashS102 1990

[7] R E Lucas Jr ldquoOn the mechanics of economic developmentrdquoJournal of Monetary Economics vol 22 no 1 pp 3ndash42 1988

[8] G M Grossman and E Helpman ldquoEndogenous innovation inthe theory of growthrdquoThe Journal of Economic Perspectives vol8 no 1 pp 23ndash44 1994

[9] P Aghion and PWHowittThe economics of growth MIT press2008

[10] L Sineviciene ldquoTax burden and economic development thecase of the European Union Countriesrdquo in EntrepreneurshipBusiness and Economics vol 2 pp 283ndash298 Springer Interna-tional Publishing Switzerland 2016

[11] J R Hines Excess Burden of Taxation Working Paper SeriesUniversity of Michigan and NBER 2007 Product Number WP2007-1

[12] J R Hines Jr ldquoThree sides of Harberger trianglesrdquo Journal ofEconomic Perspectives (JEP) vol 13 no 2 pp 167ndash188 1999

[13] A C Harberger ldquoThe measurement of wasterdquo American Eco-nomic Review vol 54 pp 58ndash76 1964

[14] A C Harberger ldquoTaxation resource allocation and welfarerdquoin The role of Direct and Indirect Taxes in the Federal RevenueSystem John FD andN J Princeton Eds pp 25ndash70 PrincetonUniversity Press 1964

[15] A Chamberlain ldquoEstimating the Tax Burden and EconomicImpact from the Proposed ldquoGang of Tenrdquo Revenue Offsetsrdquoin Fiscal Economics Policy Study 2008-08 Institute for EnergyResearch 2008 httpwwwinstituteforenergyresearchorgwp-contentuploads200809gang of 10 energy studypdf

[16] T Kbiladze ldquoTheoretical and Empirical Basis of Optimal TaxBurden in Georgiardquo International Journal of Trade Economicsand Finance vol 6 no 6 pp 314ndash317 2015

[17] G Abuselidze ldquoThe influence of optimal tax burden on eco-nomic activity and production capacityrdquo Intellectual Economicsvol 6 no 4 pp 493ndash503 2012

[18] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand Applications World Scientific Singapore 1994

[19] V Lakshmikantham and R MohapatraTheory of Fuzzy Differ-ential Equations and Inclusions Taylor amp Francis London NYUSA 2003

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] R A Aliev and R R Aliev Soft Computing and Its ApplicationsWorld Scientific Singapore New Jersey USA London UKHong Kong 2001

[22] W Pedrycz Knowledge Based Clustering from Data to Informa-tion Granules John Wiley amp Sons Hoboken NJ USA 2005

[23] I A Mayburov ldquoMethodological aspects of study of an impactof underground economy to assesment of tax burdenrdquo AuditNews vol 9 pp 50ndash62 2012 in Russian

Advances in Fuzzy Systems 3

of the underground economy A method to assess impact ofunderground economy is described in Section 4 An exampleillustrating the proposed fuzzy approach is considered inSection 5 Section 6 concludes

2 Preliminaries

Denote 1198641 is the set of fuzzy numbers defined over the realline R

Definition 1 (a fuzzy number-valued function [18 19]) Amapping 119891 119879 rarr 1198641 is referred to as a fuzzy number-valuedfunction of a real variable

Definition 2 (see [18 19]) A function 119891 119879 rarr 1198641 isreferred to as integrably bounded if there exists such real-valued function ℎ that 119909 le ℎ(119905) forall119909 isin supp119891(119905)

A definition of measurability of a fuzzy number-valuedfunction of a real variable is given in [18] In [18] it isshown that strongly measurable and integrably boundedfuzzy number-valued function is integrable

Definition 3 (an integral of a fuzzy number-valued function[18]) Let 119891 119879 rarr 1198641 An integral of 119891 on 119879 = [119886 119887]denoted as int

119879119891(119905)119889119905 or int119887

119886119891(119905)119889119905 is defined in terms of 120572-cuts

as follows

[int119879119891 (119905) 119889119905]120572 = int

119879119891120572 (119905) 119889119905 = int

119879120593 (119905) 119889119905 120593 119879

997888rarr 119877119899 minusmeasurable choice for 119891120572 forall120572 isin (0 1] (1)

Definition 4 (strongly generalized differentiability [20]) Let119891 (119886 119887) rarr 1198641 and 1199050 isin (119886 119887) We say that 119891 is stronglygeneralized differentiable at 1199050 if there exists an element1198911015840(1199050) isin 1198641 such that

(i) for all Δ119905 gt 0 sufficiently small exist119891(1199050 + Δ119905) minusℎ 119891(1199050)119891(1199050) minusℎ 119891(1199050minusΔ119905) (ie the length of diam((119891(119905))120572) increases)and the limits (in the supremummetric [18 19])

limΔ119905rarr0+

119891 (1199050 + Δ119905) minusℎ119891 (1199050)Δ119905 = lim

Δ119905rarr0+

119891 (1199050) minusℎ119891 (1199050 minus Δ119905)Δ119905

= 1198911015840 (1199050)(2)

or(ii) for all Δ119905 gt 0 sufficiently small exist119891(1199050) minusℎ 119891(1199050 +Δ119905) 119891(1199050 minus Δ119905) minusℎ 119891(1199050) (ie the length of diam((119891(119905))120572)

decreases) and the limits (in the supremummetric [18 19])

limΔ119905rarr0+

119891 (1199050) minusℎ119891 (1199050 + Δ119905)(minusΔ119905) = lim

Δ119905rarr0+

119891 (1199050 minus Δ119905) minusℎ119891 (1199050)(minusΔ119905)

= 1198911015840 (1199050) (3)

minusℎ denotes Hukuhara difference

Definition 5 (possibility measure [21 22]) Given two fuzzysets defined in the same universe of discourse 119883 a funda-mental question arises as to their similarity or proximityThere are several well-documented approaches covered in theliterature One of them concerns a possibility measure Thepossibility measure denoted by Poss(119860119883) describes a levelof overlap between two fuzzy sets and is expressed in the form

Poss (119860119883) = sup119909isin119883

[119860 (119909) and 119883 (119909)] (4)

where and is a 119905-norm Computationally we note that thepossibility measure is concerned with the determinationof the intersection between 119860 and 119883 that is followed bythe optimistic assessment of this intersection It is done bypicking up the highest values among the intersection gradesof 119860 and119883 that are taken over all elements of the universe ofdiscourse1198833 Fuzzy Model of a Tax Burden

First let us provide an explanation of important economicindicators considered in the paper

As a rule a tax burden 119906 is defined as the following ratio

119906 = 119879120585119870 (5)

where 119879 is tax revenue and 120585119870 is equity capital load takinginto account taxes

A tax burden leads to an economic system transition from119910(119905) state to 1199101(119905) state Assume that at the beginning of aconsidered year GDP depends on a value of a fixed capital

119870119894119901(119905) = 119875(119905minus1) (119870119894119901(119905minus1) minus 119877119894119901(119905minus1) minus 119860 (119905minus1)) (6)

where 119877119894119901 is investment 119860 = 120579119870 depreciation-amortizationof119870 value loaded share of core capital 120579 is amortization rate119875 is deflator

It is obvious that information related to some parametersof underground economy and economic growth in general isimpreciseThus it is more adequately to use fuzzy models fordescribing tax burden in such conditions

A tax burden 119906(119905) leads to an economic system transitionfrom fuzzy state 119910(119905) to fuzzy state 1199101(119905)

1199101 (119905) = 119906 (119905) 119910 (119905) (7)

Let us use the following terms and symbols 119905 is financial(taxation) year 119910(119905) is the income of the economic systemfrom the production and service rendered by the business1199101(119905) is posttax income of an economic entity 119906(119905) is thetax burden of the economic entity in the financial (tax) year120585 is equity capital load taking into account the taxes isthe fuzzy value of the growth rate of the economy 120578 is thefuzzy value of the underground economy weight 120591 is periodfor tax liabilities The following fuzzy equation comes fromeconomic balance conditions

int1205910[119910 (119905) minus 1199101 (119905)] 119889119905 = (1 + 120578) 120585 (8)

4 Advances in Fuzzy Systems

or

int1205910[119910 (119905) minus 119906 (119905) 119910 (119905)] 119889119905 = (1 + 120578) 120585 (9)

Now we need to analyze the form of 119910(119905) As we considerproportional economic growth model under fuzzy environ-ment dynamics of 119910 can be adequately described by thefollowing fuzzy differential equation [18ndash20] under (ii) typeof strongly generalized differentiability (Definition 4) [20]

119889119910119889119905 = 119910 (10)

It is known that the solution of (10) is

119910 = 1199100119890119905 (11)

which can be described in 120572-cuts as follows1199101205721 = 119910012057211198901205821205721 1199051199101205722 = 119910012057221198901205821205722 119905

(12)

Thus given solution (11) we can transform (9) into thefollowing integral equation

int1205910(1 minus 119906 (119905)) 119890119905119889119905 = 120585 (1 + 120578) (13)

Given the obtained fuzzymodel of tax burden we need toconsider a problem of determination of 119906(119905) that solves (13)Wewill consider determination of 119906(119905) in the class of constantfunctions 119906(119905) = 119906 119906 = const As (13) is fuzzy the equality ofthe left and the right hand sides should be considered in termsof a degree As a measure of a degree of equality we will usethe possibility measure Then the considered problem can beformulated as follows

119875(int1205910(1 minus 119906 (119905)) 119890119905119889119905 120585 (1 + 120578)) 997888rarr max (14)

st 119906 (119905) = 119906119906 = const (15)

where119875 is the possibilitymeasure In otherwords it is neededto find such a value of 119906 = const for which the possibilitymeasure of equality of the left and the right hand sides ismaximal

4 Expertsrsquo Group Opinion Based Estimationof the Impact of Underground Economy onTax Burden

Experience has shown that comprehensive indicators of aneconomy system are not quite precise due to uncertainty ofmajor factors One of the most important of these factors isthe weight of the underground economy It is of particularimportance if one takes into account comprehensive burdenfor the economyNowonder assessing of this factor is in focusof major modern surveys

00102030405060708091

091 094 097 1 103 106 109 112088

Figure 1 The fuzzy values of the left hand side (dashed curve) andthe right hand sides (solid curve) of (14)

In [23] the theoretical approaches to the definition of thehidden economy taxation base are widely analyzed specifiedinto three groups and recommended to use the volume ofGDP as the tax base However there is no comment on howto determine the specific weight of the hidden economyObviously the problem of determining both the specificweight of hidden economy and tax base comprised its taxburden is one of the pending problems

In this study the load factor of the main capital deter-mining the loaded part of the fixed capital 120585 is specifiedby the quantity 120578 As an underground economy is of latentcharacter an adequate estimation of its impact 120578 can bebased on expertsrsquo opinions described in natural language andformalized by using fuzzy sets

The opinion of expert group on the impact of changes intax legislation on tax revenue is computed by using aweightedaverage based aggregation

120578 = sum119899119894=1 119902119894 sdot 119908119894sum119899119894=1 119908119894 119894 = 1 119899 (16)

where 119902119894 is a fuzzy evaluation of 120578 provided by 119894th expert and119908119894 is a fuzzy confidence degree of 119894th expert

5 An Example

Consider the following example Let and = (10 30 50)be a triangular fuzzy number (TFN) describing fuzzyvalue of the growth rate of hypothetic economy let 120578 =(032 034 035) be a TFN describing expert group-basedcomputed by using (16) and 120585 = 0737 120591 = 1 in(13) It can be shown that the left hand side of (13) is anintegrable fuzzy number-valued function of a real variableThus we can consider a problem (14) to determine 119906 Byusing numerical techniques we have found 119906 = 015 as thesolution of (14)which provides119875(int120591

0(1minus119906(119905))119890119905119889119905 120585(1+120578)) =

096 This value of possibility measure can be consideredas sufficiently suitable from practical point of view The leftpart of (13) approximated by TFN int120591

0(1 minus 119906(119905))119890119905119889119905 =

(089 099 11) and the right part described by TFN 120585(1+120578) =(09733 09857 09982) are shown in Figure 1

Advances in Fuzzy Systems 5

6 Conclusion

In this paper we use fuzzy integral equation to model theway tax burden induces transition of state of economy Anoptimal value of tax burden is considered as a solutionto fuzzy integral equation The use of fuzzy sets helpsto model imprecise information including expertsrsquo opinionbased evaluation of an underground economy weight Anexample is used to illustrate the proposed approach In theexample a tax burden 119906 = 15 is found as a solution tothe fuzzy integral equation with possibility degree 096 Theobtained result shows validity of the proposed study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y V Balatsky ldquoA mechanism of dependence between innova-tions and economic growthrdquo Science Innovations and Educa-tion vol 2 pp 182ndash198 2007 (Russian)

[2] I A Mayburov Theory of Taxation Advanced Course UnityMoscow Russia 2011

[3] J McDermott ldquoMercantilism and modern growthrdquo Journal ofEconomic Growth vol 4 no 1 pp 55ndash80 1999

[4] RM Solow ldquoA contribution to the theory of economic growthrdquoQuarterly Journal of Economics vol 70 pp 56ndash94 1956

[5] P M Romer ldquoIncreasing returns and long-run growthrdquo Journalof Political Economy vol 94 no 5 pp 1002ndash1037 1986

[6] P M Romer ldquoEndogenous technological changerdquo Journal ofPolitical Economy vol 98 no 5 pp S71ndashS102 1990

[7] R E Lucas Jr ldquoOn the mechanics of economic developmentrdquoJournal of Monetary Economics vol 22 no 1 pp 3ndash42 1988

[8] G M Grossman and E Helpman ldquoEndogenous innovation inthe theory of growthrdquoThe Journal of Economic Perspectives vol8 no 1 pp 23ndash44 1994

[9] P Aghion and PWHowittThe economics of growth MIT press2008

[10] L Sineviciene ldquoTax burden and economic development thecase of the European Union Countriesrdquo in EntrepreneurshipBusiness and Economics vol 2 pp 283ndash298 Springer Interna-tional Publishing Switzerland 2016

[11] J R Hines Excess Burden of Taxation Working Paper SeriesUniversity of Michigan and NBER 2007 Product Number WP2007-1

[12] J R Hines Jr ldquoThree sides of Harberger trianglesrdquo Journal ofEconomic Perspectives (JEP) vol 13 no 2 pp 167ndash188 1999

[13] A C Harberger ldquoThe measurement of wasterdquo American Eco-nomic Review vol 54 pp 58ndash76 1964

[14] A C Harberger ldquoTaxation resource allocation and welfarerdquoin The role of Direct and Indirect Taxes in the Federal RevenueSystem John FD andN J Princeton Eds pp 25ndash70 PrincetonUniversity Press 1964

[15] A Chamberlain ldquoEstimating the Tax Burden and EconomicImpact from the Proposed ldquoGang of Tenrdquo Revenue Offsetsrdquoin Fiscal Economics Policy Study 2008-08 Institute for EnergyResearch 2008 httpwwwinstituteforenergyresearchorgwp-contentuploads200809gang of 10 energy studypdf

[16] T Kbiladze ldquoTheoretical and Empirical Basis of Optimal TaxBurden in Georgiardquo International Journal of Trade Economicsand Finance vol 6 no 6 pp 314ndash317 2015

[17] G Abuselidze ldquoThe influence of optimal tax burden on eco-nomic activity and production capacityrdquo Intellectual Economicsvol 6 no 4 pp 493ndash503 2012

[18] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand Applications World Scientific Singapore 1994

[19] V Lakshmikantham and R MohapatraTheory of Fuzzy Differ-ential Equations and Inclusions Taylor amp Francis London NYUSA 2003

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] R A Aliev and R R Aliev Soft Computing and Its ApplicationsWorld Scientific Singapore New Jersey USA London UKHong Kong 2001

[22] W Pedrycz Knowledge Based Clustering from Data to Informa-tion Granules John Wiley amp Sons Hoboken NJ USA 2005

[23] I A Mayburov ldquoMethodological aspects of study of an impactof underground economy to assesment of tax burdenrdquo AuditNews vol 9 pp 50ndash62 2012 in Russian

4 Advances in Fuzzy Systems

or

int1205910[119910 (119905) minus 119906 (119905) 119910 (119905)] 119889119905 = (1 + 120578) 120585 (9)

Now we need to analyze the form of 119910(119905) As we considerproportional economic growth model under fuzzy environ-ment dynamics of 119910 can be adequately described by thefollowing fuzzy differential equation [18ndash20] under (ii) typeof strongly generalized differentiability (Definition 4) [20]

119889119910119889119905 = 119910 (10)

It is known that the solution of (10) is

119910 = 1199100119890119905 (11)

which can be described in 120572-cuts as follows1199101205721 = 119910012057211198901205821205721 1199051199101205722 = 119910012057221198901205821205722 119905

(12)

Thus given solution (11) we can transform (9) into thefollowing integral equation

int1205910(1 minus 119906 (119905)) 119890119905119889119905 = 120585 (1 + 120578) (13)

Given the obtained fuzzymodel of tax burden we need toconsider a problem of determination of 119906(119905) that solves (13)Wewill consider determination of 119906(119905) in the class of constantfunctions 119906(119905) = 119906 119906 = const As (13) is fuzzy the equality ofthe left and the right hand sides should be considered in termsof a degree As a measure of a degree of equality we will usethe possibility measure Then the considered problem can beformulated as follows

119875(int1205910(1 minus 119906 (119905)) 119890119905119889119905 120585 (1 + 120578)) 997888rarr max (14)

st 119906 (119905) = 119906119906 = const (15)

where119875 is the possibilitymeasure In otherwords it is neededto find such a value of 119906 = const for which the possibilitymeasure of equality of the left and the right hand sides ismaximal

4 Expertsrsquo Group Opinion Based Estimationof the Impact of Underground Economy onTax Burden

Experience has shown that comprehensive indicators of aneconomy system are not quite precise due to uncertainty ofmajor factors One of the most important of these factors isthe weight of the underground economy It is of particularimportance if one takes into account comprehensive burdenfor the economyNowonder assessing of this factor is in focusof major modern surveys

00102030405060708091

091 094 097 1 103 106 109 112088

Figure 1 The fuzzy values of the left hand side (dashed curve) andthe right hand sides (solid curve) of (14)

In [23] the theoretical approaches to the definition of thehidden economy taxation base are widely analyzed specifiedinto three groups and recommended to use the volume ofGDP as the tax base However there is no comment on howto determine the specific weight of the hidden economyObviously the problem of determining both the specificweight of hidden economy and tax base comprised its taxburden is one of the pending problems

In this study the load factor of the main capital deter-mining the loaded part of the fixed capital 120585 is specifiedby the quantity 120578 As an underground economy is of latentcharacter an adequate estimation of its impact 120578 can bebased on expertsrsquo opinions described in natural language andformalized by using fuzzy sets

The opinion of expert group on the impact of changes intax legislation on tax revenue is computed by using aweightedaverage based aggregation

120578 = sum119899119894=1 119902119894 sdot 119908119894sum119899119894=1 119908119894 119894 = 1 119899 (16)

where 119902119894 is a fuzzy evaluation of 120578 provided by 119894th expert and119908119894 is a fuzzy confidence degree of 119894th expert

5 An Example

Consider the following example Let and = (10 30 50)be a triangular fuzzy number (TFN) describing fuzzyvalue of the growth rate of hypothetic economy let 120578 =(032 034 035) be a TFN describing expert group-basedcomputed by using (16) and 120585 = 0737 120591 = 1 in(13) It can be shown that the left hand side of (13) is anintegrable fuzzy number-valued function of a real variableThus we can consider a problem (14) to determine 119906 Byusing numerical techniques we have found 119906 = 015 as thesolution of (14)which provides119875(int120591

0(1minus119906(119905))119890119905119889119905 120585(1+120578)) =

096 This value of possibility measure can be consideredas sufficiently suitable from practical point of view The leftpart of (13) approximated by TFN int120591

0(1 minus 119906(119905))119890119905119889119905 =

(089 099 11) and the right part described by TFN 120585(1+120578) =(09733 09857 09982) are shown in Figure 1

Advances in Fuzzy Systems 5

6 Conclusion

In this paper we use fuzzy integral equation to model theway tax burden induces transition of state of economy Anoptimal value of tax burden is considered as a solutionto fuzzy integral equation The use of fuzzy sets helpsto model imprecise information including expertsrsquo opinionbased evaluation of an underground economy weight Anexample is used to illustrate the proposed approach In theexample a tax burden 119906 = 15 is found as a solution tothe fuzzy integral equation with possibility degree 096 Theobtained result shows validity of the proposed study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y V Balatsky ldquoA mechanism of dependence between innova-tions and economic growthrdquo Science Innovations and Educa-tion vol 2 pp 182ndash198 2007 (Russian)

[2] I A Mayburov Theory of Taxation Advanced Course UnityMoscow Russia 2011

[3] J McDermott ldquoMercantilism and modern growthrdquo Journal ofEconomic Growth vol 4 no 1 pp 55ndash80 1999

[4] RM Solow ldquoA contribution to the theory of economic growthrdquoQuarterly Journal of Economics vol 70 pp 56ndash94 1956

[5] P M Romer ldquoIncreasing returns and long-run growthrdquo Journalof Political Economy vol 94 no 5 pp 1002ndash1037 1986

[6] P M Romer ldquoEndogenous technological changerdquo Journal ofPolitical Economy vol 98 no 5 pp S71ndashS102 1990

[7] R E Lucas Jr ldquoOn the mechanics of economic developmentrdquoJournal of Monetary Economics vol 22 no 1 pp 3ndash42 1988

[8] G M Grossman and E Helpman ldquoEndogenous innovation inthe theory of growthrdquoThe Journal of Economic Perspectives vol8 no 1 pp 23ndash44 1994

[9] P Aghion and PWHowittThe economics of growth MIT press2008

[10] L Sineviciene ldquoTax burden and economic development thecase of the European Union Countriesrdquo in EntrepreneurshipBusiness and Economics vol 2 pp 283ndash298 Springer Interna-tional Publishing Switzerland 2016

[11] J R Hines Excess Burden of Taxation Working Paper SeriesUniversity of Michigan and NBER 2007 Product Number WP2007-1

[12] J R Hines Jr ldquoThree sides of Harberger trianglesrdquo Journal ofEconomic Perspectives (JEP) vol 13 no 2 pp 167ndash188 1999

[13] A C Harberger ldquoThe measurement of wasterdquo American Eco-nomic Review vol 54 pp 58ndash76 1964

[14] A C Harberger ldquoTaxation resource allocation and welfarerdquoin The role of Direct and Indirect Taxes in the Federal RevenueSystem John FD andN J Princeton Eds pp 25ndash70 PrincetonUniversity Press 1964

[15] A Chamberlain ldquoEstimating the Tax Burden and EconomicImpact from the Proposed ldquoGang of Tenrdquo Revenue Offsetsrdquoin Fiscal Economics Policy Study 2008-08 Institute for EnergyResearch 2008 httpwwwinstituteforenergyresearchorgwp-contentuploads200809gang of 10 energy studypdf

[16] T Kbiladze ldquoTheoretical and Empirical Basis of Optimal TaxBurden in Georgiardquo International Journal of Trade Economicsand Finance vol 6 no 6 pp 314ndash317 2015

[17] G Abuselidze ldquoThe influence of optimal tax burden on eco-nomic activity and production capacityrdquo Intellectual Economicsvol 6 no 4 pp 493ndash503 2012

[18] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand Applications World Scientific Singapore 1994

[19] V Lakshmikantham and R MohapatraTheory of Fuzzy Differ-ential Equations and Inclusions Taylor amp Francis London NYUSA 2003

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] R A Aliev and R R Aliev Soft Computing and Its ApplicationsWorld Scientific Singapore New Jersey USA London UKHong Kong 2001

[22] W Pedrycz Knowledge Based Clustering from Data to Informa-tion Granules John Wiley amp Sons Hoboken NJ USA 2005

[23] I A Mayburov ldquoMethodological aspects of study of an impactof underground economy to assesment of tax burdenrdquo AuditNews vol 9 pp 50ndash62 2012 in Russian

Advances in Fuzzy Systems 5

6 Conclusion

In this paper we use fuzzy integral equation to model theway tax burden induces transition of state of economy Anoptimal value of tax burden is considered as a solutionto fuzzy integral equation The use of fuzzy sets helpsto model imprecise information including expertsrsquo opinionbased evaluation of an underground economy weight Anexample is used to illustrate the proposed approach In theexample a tax burden 119906 = 15 is found as a solution tothe fuzzy integral equation with possibility degree 096 Theobtained result shows validity of the proposed study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y V Balatsky ldquoA mechanism of dependence between innova-tions and economic growthrdquo Science Innovations and Educa-tion vol 2 pp 182ndash198 2007 (Russian)

[2] I A Mayburov Theory of Taxation Advanced Course UnityMoscow Russia 2011

[3] J McDermott ldquoMercantilism and modern growthrdquo Journal ofEconomic Growth vol 4 no 1 pp 55ndash80 1999

[4] RM Solow ldquoA contribution to the theory of economic growthrdquoQuarterly Journal of Economics vol 70 pp 56ndash94 1956

[5] P M Romer ldquoIncreasing returns and long-run growthrdquo Journalof Political Economy vol 94 no 5 pp 1002ndash1037 1986

[6] P M Romer ldquoEndogenous technological changerdquo Journal ofPolitical Economy vol 98 no 5 pp S71ndashS102 1990

[7] R E Lucas Jr ldquoOn the mechanics of economic developmentrdquoJournal of Monetary Economics vol 22 no 1 pp 3ndash42 1988

[8] G M Grossman and E Helpman ldquoEndogenous innovation inthe theory of growthrdquoThe Journal of Economic Perspectives vol8 no 1 pp 23ndash44 1994

[9] P Aghion and PWHowittThe economics of growth MIT press2008

[10] L Sineviciene ldquoTax burden and economic development thecase of the European Union Countriesrdquo in EntrepreneurshipBusiness and Economics vol 2 pp 283ndash298 Springer Interna-tional Publishing Switzerland 2016

[11] J R Hines Excess Burden of Taxation Working Paper SeriesUniversity of Michigan and NBER 2007 Product Number WP2007-1

[12] J R Hines Jr ldquoThree sides of Harberger trianglesrdquo Journal ofEconomic Perspectives (JEP) vol 13 no 2 pp 167ndash188 1999

[13] A C Harberger ldquoThe measurement of wasterdquo American Eco-nomic Review vol 54 pp 58ndash76 1964

[14] A C Harberger ldquoTaxation resource allocation and welfarerdquoin The role of Direct and Indirect Taxes in the Federal RevenueSystem John FD andN J Princeton Eds pp 25ndash70 PrincetonUniversity Press 1964

[15] A Chamberlain ldquoEstimating the Tax Burden and EconomicImpact from the Proposed ldquoGang of Tenrdquo Revenue Offsetsrdquoin Fiscal Economics Policy Study 2008-08 Institute for EnergyResearch 2008 httpwwwinstituteforenergyresearchorgwp-contentuploads200809gang of 10 energy studypdf

[16] T Kbiladze ldquoTheoretical and Empirical Basis of Optimal TaxBurden in Georgiardquo International Journal of Trade Economicsand Finance vol 6 no 6 pp 314ndash317 2015

[17] G Abuselidze ldquoThe influence of optimal tax burden on eco-nomic activity and production capacityrdquo Intellectual Economicsvol 6 no 4 pp 493ndash503 2012

[18] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand Applications World Scientific Singapore 1994

[19] V Lakshmikantham and R MohapatraTheory of Fuzzy Differ-ential Equations and Inclusions Taylor amp Francis London NYUSA 2003

[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[21] R A Aliev and R R Aliev Soft Computing and Its ApplicationsWorld Scientific Singapore New Jersey USA London UKHong Kong 2001

[22] W Pedrycz Knowledge Based Clustering from Data to Informa-tion Granules John Wiley amp Sons Hoboken NJ USA 2005

[23] I A Mayburov ldquoMethodological aspects of study of an impactof underground economy to assesment of tax burdenrdquo AuditNews vol 9 pp 50ndash62 2012 in Russian

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