research article real national income average growth rate

Research ArticleReal National Income Average Growth Rate A Novel EconomicGrowth and Social Fair Evaluation Index

Chien Wei Wu1 and Wei Zhan Hung2

1Department of Economics National Chi Nan University No 1 University Road Puli Nantou County 54561 Taiwan2Department of International Tourism and Recreation Hsing-Kuo University No 600 Section 3 Taijiang BoulevardAnnan District Tainan Taiwan

Correspondence should be addressed to Wei Zhan Hung steady 2006hotmailcom

Received 4 September 2014 Accepted 12 December 2014

Academic Editor Joao Ricardo Faria

Copyright copy 2015 C W Wu and W Z Hung This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The purpose of this study is to propose a new economic index namely real national income average growth rate (RNIAGR)whichmeasures the performance of economic growth with consideration for income distributionThis study also develops anothernew economic index called five-scale real national income average growth rate (FSRNIAGR) which simplifies the calculation ofRNIAGRThe merits of these new indexes are discussed to justify their efficacy This paper also justifies the use of proposed indexby showing that this index can actually measure the ordering of social welfare To highlight the difference between this new indexand the traditional ones this paper compares the index with real economic growth rate using the data of Taiwan In addition thispaper shows that when the real growth stagnates or even declines this new index indicates that income distribution deteriorates

1 Introduction

In recent years an immense income disparity has become acommon phenomenon across borders For example accord-ing to a report published by the Organization for EconomicCooperation and Development (OECD) in 2011 the averageincome of the richest 10 of the population is about ninetimes that of the poorest 10 across theOECD up from seventimes in 1985 Hence income distribution has been a centralconcern for both academics and policy makers in developedcountries as well as developing countries As pointed out byRobert Shiller one of the winners of the Nobel Prize forEconomics in 2013 ldquoThemost important problem that we arefacing now today I think is rising inequality in the UnitedStates and elsewhere in the worldrdquo (The Huffington Post Oct15 2013)

A number of economic indexes are commonly usedto measure economic growth and income distribution Forexample real gross domestic product is used to measureeconomic growth Gini coefficient Lorenz curve and theratio of income share of highest 20 to that of lowest 20

measure income distribution However under certain cir-cumstances these two objectives (growth and distribution)are conflicting to each other For example if policy makersfocus on pursuing economic growth this may deteriorateuneven distribution (income gap) By contrast if they focuson decreasing uneven distribution this may lower the workincentives of high ability workers

For policy makers a single index that incorporatesgrowth as well as distribution may help to immediately crys-tallize the efficacy of certain policies Despite the prevalenceof many indexes for income inequality none of the indexesfor growth can to some extent reflect the degree of incomeinequalities The lack of such index necessitates the study ofbetter indexes that consider growth as well as distributionsimultaneously To this end this study intends to proposea new index Real National Income Average Growth Rate(hereafter RNIAGR) We justify the use of this index byshowing that this index can actually measure the ordering ofsocial welfare In addition we also show that when the realgrowth stagnates or even declines this index indicates thatincome distribution deteriorates

Hindawi Publishing CorporationEconomics Research InternationalVolume 2015 Article ID 678927 7 pageshttpdxdoiorg1011552015678927

2 Economics Research International

The reminder of this paper is organized as followsSection 2 reviews major economic growth indexes andincome distribution indexes in the existing literatureSection 3 defines the concept of real national income averagegrowth rate and Section 4 further discusses the advantages ofusing this new index In Section 5 using the data in Taiwanwe compare this new index with the usual real economicgrowth rate Finally Section 6 concludes this paperwith someremarks

2 Economic Index

21 Measuring Economic Growth Index The main economicindex for measuring economic growth is real economicgrowth rate The formula of the real economic growth rateis defined as follows

( Real GDP119905

Real GDP119905minus1

) minus 1 (1)

Real Economic Growth Rate119905represents real eco-

nomic growth rate of 119905th yearReal GDP

119905represents real gross domestic product of

119905th yearReal GDP

119905minus1represents real gross domestic product

of 119905 minus 1th year

Gross domestic product can be explained as the powerof the country Positive real economic growth rate means thepower of the country is increasing It seems every countryaims to promote their real economic growth rate

Real economic growth rate is calculated as the totalincome of all nationals in the last year divided by the totalnet increase income of all nationals in this year It can revealthe macroeconomic level of the entire nation but it does notdisplay the information of the income growth rate of eachindividual nationHowever promoting real economic growthrate usually leads to a side effect unequal income distribution[1 2]

22 Measuring Income Distribution Index The main eco-nomic index for measuring income distribution includesLorenz curve Gini coefficient and ratio of income share ofhighest 20 to that of lowest 20

Lorenz curve is an income aggregation curve whichaggregates income from the lowest income national to thehighest income national [3]

The Gini coefficient is a measuring index of statisticaldispersion formulated by Corrado Gini [4] Gini coefficientis usually defined mathematically based on the Lorenz curveGini coefficient ranges from 0 to 1 A low Gini coefficientindicates a more equal distribution Gini coefficient = 0 rep-resents complete equality Higher Gini coefficients indicatemore unequal distribution Gini coefficient = 1 representscomplete inequality Although Gini coefficient can explainthe unequal degree of income distribution it is useless to useGini coefficient to express the financial net increase incomedistribute of each national in each year

As implied by the name ratio of income share of highest20 to that of lowest 20means the total income of nationalswhose income rank is in lowest 20 divides the total incomeof nationals whose income rank is in highest 20 [5] Thedrawback of ratio of income share of highest 20 to that oflowest 20 is that it does not consider the incomedistributionof nationals whose income is in the middle

3 Real National Income Average Growth Rate

To consider economic growth and incomedistribution simul-taneous this research provides a new indexmdashreal nationalincome average growth rateThe concept of RNIAGR is aver-aged real income growth of each individual national Unlikereal economic growth rate that calculates the economicgrowth rate based on the total income of all nationals inprevious year RNIAGR is calculated from individual incomegrowth rate based on individual income in the previous yearand average of all of individual income growth rate RNIAGRcan reflect the average income growth status of individualnational This formula supposes that the volume of nationalsin 119905th year and in 119905minus1th year in the sameCountry is the same

RNIAGR119905= 1119901 lowast119901

sum119895=1

( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1 (2)

RNIAGR119905represents real national income average

growth rate in 119905th year119868119895119905represents the income of 119895th national in 119905th year

119868119895119905minus1

represents the income of 119895th national in 119905 minus 1thyear119865119905represents the inflation rate in 119905th year

119875 represents the volume of nationals in the country

Using Figure 1 as example 119909-axis represents the rank ofincome from low income national to high income nationalPoint 119860 represents the lowest income national and point 119861represents the highest incomenational Frompoint119860 to point119861 all of the points can reflect an income of one national

If Area119860119861119862 represents the total income of all nationals inlast year Area 119860119862119863 represents the net increase income of allnationals in this year So nominal economic growth rate canbe represented as Area 119860119861119862 divides Area 119860119862119863 From point119860 to point 119861 all of the points can reflect an income of onenational Every national uses the individual income as a baseto calculate the individual income growth rate And we canobtain nominal national income average growth by averagingthe individual income growth rate For example point 119894 thatis between point 119860 and point 119861 represents 119894th national Theincome of 119894th national in last year is line 119894119895 The net increaseincome of 119894th national in this year is line 119895119896 The individualincome growth rate of 119894th national is line 119894119895 divide line 119895119896In order to reflect the real income status of each nationalnominal national income average growth needs to be dividedby inflation rate of this year to acquire real national incomeaverage growth rate of this year

Economics Research International 3

A B

C

D

i

j

k

Citizen

Income

x-axis

Figure 1 National income distribution diagram Notice the orderof the citizens at axis ldquo119909rdquo can (and probably will) change from oneyear to another

Because it is not easy and is too complex to acquireincome information of each individual this study presentsother new indexesmdashfive-scale real national income averagegrowth rate (FSRNIAGR) which can roughly represent realnational income average growth rate FSRNIAGR can reducethe calculation complexity without losing the meaning ofRNIAGR First we divide the entire nationals into 5 scalesAnd then we aggregate the income of each national in eachscale in the last year and aggregate the net increase incomeof each national in each scale in this year In each scalethe aggregated income of each national in these scales inthe last year divides the aggregated net increase income ofeach national in these scales in this year as nominal nationalincome average growth rate in each scale Finally nominalfive-scale national income average growth rates are acquiredby averaging nominal national income average growth ratein each scale In order to reflect the real income status of eachnational nominal five-scale national income average growthneeds to be divided by inflation rate of this year to acquire realfive-scale national income average growth rate of this year(Figure 2)

FSRNIAGR119905

= (5

sum119894=1

( sum(1199011199055)lowast119894119895=(1199011199055)lowast(119894minus1)+1

119868119895119905

sum(119901119905minus15)lowast119894119895=(119901119905minus15)lowast(119894minus1)+1

119868119895119905minus1

)

lowast(119875119905minus1119875119905)(1 + 119865119905) ) times (5)minus1 minus 1

(3)

FSRNIAGR119905

represents five-scale real nationalincome average growth rate of 119905th year119868119895119905represents the ordered income of 119895th national in

119905th year119868119895119905minus1

represents the ordered income of 119895th national in119905 minus 1th year

A B

C

D

s1 s2 s3 s4

c1c2

c3

c4

d1

d2

d3

d4

Citizen

Income

x-axis

Figure 2 Five-scale national income distribution diagram

119865119905represents the inflation rate in 119905th year

119901119905represents the volume of nationals in the Country

in 119905th year119901119905minus1

represents the volumeof nationals in theCountryin 119905 minus 1th year(1199011199055)lowast (119894 minus 1) represents the lower bound of national

in 119894th scale in 119905th year(119901119905minus15)lowast(119894minus1) represents the lower bound of national

in 119894th scale in 119905 minus 1th year(1199011199055) lowast 119894 represents the upper bound of national in

119894th scale in 119905th year(119901119905minus15) lowast 119894 represents the upper bound of national in

119894th scale in 119905 minus 1th year

Because the volume of nationals in 119905th year and in 119905 minus 1thyear in the same Country is different FSRNIAGR

119905must be

adjusted by (119875119905minus1119875119905) for acquiring the rational growth rate

In reality the variance of income for each nationalbetween each year is not always positive and the nationals ofthe country are not usually adjusted their income accordingto their income in the previous year So 119868

119895119905minus1represents the

income of national whose financial rank is 119895th in 119905 minus 1th yearand 119868119895119905represents the income of national whose financial rank

is 119895th in 119905th yearUsing Figure 2 as an example 119909-axis represents the rank

of income from low income national to high income nationalPoint 119860 represents the lowest income national and point119861 represents the highest income national Line 119860119904

1 11990411199042

11990421199043 11990431199044 1199044119861 represents national sets The national whose

income is classified as the same income scale is in thesame national set Line 119860119904

1represents the nationals whose

income is ranking in lowest 20 Line 11990411199042represents the

nationals whose income is ranking between lowest 20and lowest 40 Line 119904


income is ranking between lowest 40 and lowest 60Line 119904

31199044represents the nationals whose income is ranking

between highest 20 and highest 40 Line 1199044119861 represents

the nationals whose income is ranking in highest 20


Area 11986011990411198881represents the total income of the nationals

whose income is ranking in lowest 20 in the last year Area11986011990411198891represents the total income of the nationals whose

income is ranking in lowest 20 in this year Area 1199041119888111988821199042

represents the total income of the nationals whose income isranking between lowest 20 and lowest 40 in the last yearArea 119904

1119889111988921199042represents the total income of the nationals

whose income is ranking between lowest 20 and lowest 40in this year and suchlike

So nominal income growth rate of nationals whoseincome is ranking in lowest 20 can be represented as Area11986011990411198881divides Area119860119904

11198891minus1 Nominal income growth rate of

nationals whose income is ranking between lowest 20 andlowest 40 can be represented as Area 119904

1119888111988821199042divides Area

1199041119889111988921199042minus 1 and the like

Five-scale real national income average growth rate (FSR-NIAGR) of 119905th year can be presented as follows

FSRNIAGR119905

= ((119860 (11986011990411198891)119860 (11986011990411198881) +

119860 (1199041119889111988921199042)

119860 (1199041119888111988821199042) + 119860 (119904

2119889211988931199043)

119860 (1199042119888211988831199043)

+119860 (1199043119889311988941199044)119860 (1199043119888311988841199044) + 119860 (119904

41198894119863119861)

119860 (11990441198884119862119861) ) lowast (119875

119905minus1119875119905)

(1 + 119865119905) )

times (5)minus1 minus 1(4)

119860(11986011990411198891) represents the total income of nationals

whose income is ranking in lowest 20 in 119905th year119860(11986011990411198881) represents the total income of nationals

whose income is ranking in lowest 20 in 119905minus1th year119860(1199041119889111988921199042) represents the total income of nationals

whose income is ranking between lowest 20 andlowest 40 in 119905th year119860(1199041119888111988821199042) represents the total income of nationals

whose income is ranking between lowest 20 andlowest 40 in 119905 minus 1th year119860(1199042119889211988931199043) represents the total income of nationals



whose income is ranking highest 20 and highest40 in 119905th year119860(1199043119888311988841199044) represents the total income of nationals

whose income is ranking between highest 20 andhighest 40 in 119905 minus 1th year119860(11990441198894119863119861) represents the total income of nationals

whose income is ranking in highest 20 in 119905th year119860(11990441198884119862119861) represents the total income of nationals

whose income is ranking in highest 20 in 119905 minus 1thyear

25000

20000

15000

10000

5000

0Low

nationalincome

High nationalincome

Low nationalincome

High nationalincome

Low nationalincome

High nationalincome

Figure 3 National income example

In order to reflect the real income status of each nationalfive-scale nominal national income average growth rate needsto be divided by inflation rate of this year to acquire five-scalereal national income average growth rate of this year

4 The Merits of the Real National IncomeAverage Growth Rate

The strong point of real national income average growth rateis that it can modify the drawback of real economic growthrate which did not consider income distribution

For example we suppose that there exists a country whopossesses two nationalsThe income of one national is 20000US dollars in last year (blue color area) and the income of theother national is 10000US dollars in last year (red color area)Suppose that gross domestic product increases 3000 US thisyear The inflation rate of this year is 2

So the real economic growth rate in this country is((20000 + 10000 + 3000)(20000 + 10000))(1 + 2) minus1 = 78

Case 1 (all additional income is acquired by the rich individ-ual) If all the economic profit is acquired by higher nationalthe real national income average growth rate is (05 lowast(1000010000) + 05 lowast (2300020000))(1 + 2) minus 1 = 54

Case 2 (all additional income is acquired by the poorindividual) If all the economic profit is acquired by lowernational the real national income average growth rate is 05 lowast(1300010000) + 05 lowast (2000020000) minus 1 = 127

Case 3 (all additional income is acquired by each individualand the additional income is distributed according to theprevious income of each individual) If economic profit isdistributed according to the income of each national lastyear the real national income average growth rate is (05 lowast(1100010000) + 05 lowast (2200020000))(1 + 2) minus 1 = 78(Refer to Figure 3)

When the level of unequal income distribution is lowerreal national income average growth rate (127) is largerthan real economic growth rate (78) Otherwise realnational income average growth rate (54) is larger thanreal economic growth rate (78) when the level of unequal


Table 1 The relation of real economic growth rate real nationalincome average growth rate and income distribution

Real economic growth rateIncrease Decrease

The level ofunequalincomedistribution

Increase

RNIAGR isincreasing and

adjusted to lowerdown

RNIAGR isdecreasing andadjusted to lower

down more

Decrease

RNIAGR isincreasing andadjusted to bepromoted

RNIAGR isdecreasing andadjusted to

promote a few

income distribution is rising So government will get a betterperformance in real national income average growth rate(relative to real economic growth rate) when governmentpolicy can promote the income of lower income nationallargely

Real national income average growth rate not only can bepositive in relation to real economic growth rate but also canadjust real economic growth rate according to the conditionof the level of unequal income distribution For the relationof real economic growth rate real national income averagegrowth rate and income distribution refer to Table 1

5 Social Welfare Function

To give further rationale for this new index RNIAGR we setup a simplemodel to illustrate the relationship of real nationalincome average growth rate and some total social welfarefunction

We assume that each national has the same increasingutility function of income 119880(119868) = ln(119868) with a diminishingmarginal utility Further we assume that the total socialwelfare function is simply the summation of all nationalsrsquoutility function sum119901

119894=1119880(119868119894119905) where (119868

1119905 1198682119905 119868

119901119905) is the

income distribution of all nationals at time 119905We consider an income distribution 119868

119905minus1= (1198681119905minus1

1198682119905minus1

119868119901119905minus1

) at time 119905minus1 and two possible income distributions119868119905= (1198681119905 1198682119905 119868

119901119905) and 1198681015840

119905= (11986810158401119905 11986810158402119905 1198681015840

119901119905) at time 119905 We

also assume that the difference of income between time 119905 andtime 119905minus1 is relatively small that is 119868

119894119905minus119868119894119905minus1

≪ 119868119894119905minus1

and 1198681015840119894119905minus

119868119894119905minus1

≪ 119868119894119905minus1

for 119894 = 1 2 119901 Let Δ119880119895119905equiv 119880(119868

119895119905) minus 119880(119868

119895119905minus1)

andΔ1198801015840119895119905equiv 119880(1198681015840

119895119905)minus119880(119868

119895119905minus1)Therefore the difference of the

social welfare between two possible incomes is

sum119880(119868119895119905) minussum119880(1198681015840

119895119905)

= sum[119880 (119868119895119905minus1

) + Δ119880119894119905] minussum[119880 (119868

119895119905minus1) + Δ1015840

119895119905]

= sumΔ119880119895119905minussumΔ1198801015840

119895119905

(5)

For a small change in 119868 we might take (120597120597119868)119880(sdot)119889119868 as theapproximation to the resulting change in 119880 knowing thatthe smaller the 119889119868 the better the approximation There-fore sumΔ119880

119895119905asymp sum(120597120597119868)119880(119868

119895119905minus1) lowast 119889119868

119895and sumΔ1198801015840

119895119905asymp

sum(120597120597119868)119880(119868119895119905minus1

) lowast 1198891198681015840119895 where 119889119868

119895= 119868119895119905

minus 119868119895119905minus1

and

1198891198681015840119895

= 1198681015840119895119905

minus 119868119895119905minus1

We then define an approximation ofsum119880(119868

119895119905) minus sum119880(1198681015840

119895119905) to be

[sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886

equiv sum 120597120597119868119880 (119868

119895119905minus119894) lowast 119889119868 minussum 120597

120597119868119880 (119868119895119905minus1

) 1198891198681015840119895

= sum 119889119868119895minus 1198891198681015840119895

119868119895119905minus1

(6)

We are now ready to state the following proposition

Proposition 1 Consider the following

RNIAGR (119868119905) minus RNIAGR (1198681015840

119905)

ge 0 lt==gt [sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886 ge 0

(7)

Proof Consider the following


119905)

= 1119875 sum( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

minus 1119875 sum( 1198681015840

119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

= 1119875 (1 + 119865

119905) (sum

119868119895119905

119868119895119905minus1

minussum 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119868119895119905minus 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

(119868119895119905minus 119868119895119905minus1

) minus (1198681015840119895119905minus 119868119895119905minus1

)119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119889119868119895minus 1198891198681015840119895

119868119895119905minus1

)

(8)

Since both 119875 and (1 + 119865119905) are bigger than zero it is obvious

that RNIAGR(119868119905) minus RNIAGR(1198681015840

119905) ge 0 lt==gt [sum119880(119868

119895119905) minus

sum119880(1198681015840119895119905)]119886 ge 0

With the above proposition we could use the RNIAGR asan approximation of social welfare

6 Examples

To explain the usefulness of real national income averagegrowth rate this study uses Taiwanrsquos data as an example tocompare the result of real economic growth rate and realnational income average growth rate The data about realGDP nominal GDP five-scale disposable incomes consumerprice index and Gini index are collected from Taiwanrsquosgovernment [6] Inflation rate is represented by consumer


Table 2 The comparison result of real economic growth rate and real national income average growth rate in Taiwan

Year

National income growth rate based on different equal divisions of recipients

Real nationalincomeaveragegrowth rate

Real economicgrowth rate Gini IndexThe national

whose incomerank is inlowest 20

The nationalwhose income

rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween

highest 20and highest

40

The nationalwhose incomerank is in

highest 20

1995 624 150 minus077 minus157 minus017 105 003 03171996 minus362 minus346 minus332 minus220 minus313 minus315 minus303 03171997 444 216 122 050 146 195 149 03201998 265 168 025 078 146 136 121 03241999 060 197 196 093 048 119 103 03252000 173 179 027 078 minus072 077 028 03262001 minus893 minus833 minus570 minus363 219 minus488 minus264 03502002 141 053 minus034 minus133 minus102 minus015 minus062 03452003 371 294 196 166 024 210 142 03432004 157 115 229 216 139 171 170 03382005 minus001 051 minus117 minus121 minus127 minus063 minus093 03402006 minus261 minus250 minus120 minus023 minus159 minus163 minus141 03392007 047 082 minus025 minus091 059 014 013 03402008 minus083 minus448 minus449 minus271 minus178 minus285 minus270 03412009 minus878 minus591 minus504 minus604 minus602 minus636 minus606 03452010 481 415 403 394 243 387 342 03422011 minus358 minus279 minus356 minus344 minus135 minus294 minus253 0342

price index The unit of data is 1 year The range of data isbetween 1995 and 2011 To compare the results of real eco-nomic growth rate and real national income average growthrate refer to Table 2 and Figure 4

As indicated by Taiwanrsquos economic data the real eco-nomic growth rate had been lower than 2 since 1995 exceptfor the 364 in 2010 Seven out of sixteen years had seennegative growthThe rates of growth in 2001 2008 2009 and2011 were all worse than minus2

Real national income average growth rate can adjust realeconomic growth rate based on the real growth situation ofcitizen group For example the real economic growth rate ofTaiwan in 1995 is only 003 Real income growth rate of thenational whose income is ranking in lowest 20 is 624 realincome growth rate of the national whose income is rankingbetween lowest 20 and lowest 40 is 150 Real nationalincome average growth rate is 105 Although the realeconomic growth rate does not increase real national incomeaverage growth rate also can acquire high performance if theunequal income distribution is improved

In 2001 the real economic growth rate was minus264 forTaiwan However the negative growth was more than 35for individuals at lower 80 income brackets and the indi-viduals at lower 40 income brackets saw a negative growthamounting to more than 8 But people in the top 20 sawtheir income growing by 219 Not only the real economicgrowth decrease but also the unequal income distribution got

worse so real national income average growth rate is minus488which is lower than real economic growth rate The Giniindex also got worse in this year (0350 gt 0326) (the higherGini index the more unequal in income distribution)

In 2010 after the poor performance in 2009 the realeconomic growth rate was 342 for Taiwan Because thepositive growth was higher than 342 for individuals atlower 80 income brackets and people in the top 20 onlysaw their income growing by 243 real national incomeaverage growth rate is 387 which is higher than realeconomic growth rate (342) In reality the Gini index alsogot better in this year (0342 lt 0345) The economic wasperforming well in this year because not only the economicgot better but also the unequal income distribution had beenimproved

The Gini index is defined based on the concept of stockbecause this index is determined according to the long termincome ability of each national The real national incomeaverage growth rate is developed by using the concept offlow because this index is generated by applying the data ofindividual national income in this year So the real nationalincome average growth rate reflects the practical phenomenaof economic more frequently than Gini index For examplethe unequal income distribution got worse quickly in Taiwanin 2001 so Gini index increased from 0326 to 0350 ButGini index decreased slowly in 2002sim2004 when the unequalincome distribution had been improved


800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011

Real income growth rate of the national whose incomeis ranking in lowest 20Real income growth rate of the national whose incomeis ranking between lowest 20 and lowest 40Real income growth rate of the national whose incomeis ranking between lowest 40 and lowest 60

Real income growth rate of the national whose incomeis ranking between highest 20 and highest 40Real income growth rate of the national whose incomeis ranking in highest 20

Real national income average growth rate

Real economic growth rate

()

Figure 4 The comparison result of real economic growth rate andreal national income average growth rate in Taiwan

There is an interesting phenomenon that occurred inTaiwan real national income average growth rate usuallyperformed better than real economic growth rate when thereal economic growth rate was positive On the other handreal national income average growth rate usually performsworse than real economic growth rate when the real eco-nomic growth rate is negative

7 Conclusion and Future Research

In this study we develop a new economic indexmdashRNIAGRfor describing economic situation which considers economicgrowth and income distribution simultaneous Proposedindex not only can reflect real economic growth rate butalso can adjust real economic growth rate according to thecondition of the level of unequal income distribution Inorder to lower down calculation complexity this researchalso develops an economic indexmdashFSRNIAGR which cansimplify calculation process without losing the meaning ofRNIAGR

The reason of choosing FSRNIAGR is that the indexof FSRNIAGR can be generated by five-scale disposableincome which is easy to be acquired in the website in eachgovernment

In the future some economic numerical data will becollected and the economic performance comparison ofeach country based on proposed index will be discussedThe sensitivity analysis in relation to the arbitrary number

of classes which is used to classify the group of incomedistribution will be done by the simulation according to thepractical income distribution of each country

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Kanbur ldquoChapter 13 Income distribution and developmentrdquoHandbook of Income Distribution vol 1 pp 791ndash841 2000

[2] R Kanbur ldquoEconomic policy distribution and poverty thenature of disagreementsrdquoWorld Development vol 29 no 6 pp1083ndash1094 2001

[3] V Fakoor M B Ghalibaf and H A Azarnoosh ldquoAsymptoticbehaviors of the Lorenz curve and Gini index in sampling froma length-biased distributionrdquo Statistics and Probability Lettersvol 81 no 9 pp 1425ndash1435 2011

[4] BMilanovic ldquoA simpleway to calculate theGini coefficient andsome implicationsrdquo Economics Letters vol 56 no 1 pp 45ndash491997

[5] R Anderton Macroeconomic Performance in a GlobalisingEconomy Cambridge University Press Cambridge UK 2010

[6] Dorectorate-General of Budget Accounting and Statistics ofExecutive Yuan Taiwan httpwwwdgbasgovtw

Submit your manuscripts athttpwwwhindawicom

Child Development Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Education Research International


Biomedical EducationJournal of



Psychiatry Journal

ArchaeologyJournal of



AnthropologyJournal of


Research and TreatmentSchizophrenia


Urban Studies Research

Population ResearchInternational Journal of


CriminologyJournal of


Aging ResearchJournal of



NursingResearch and Practice

Current Gerontologyamp Geriatrics Research

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Sleep DisordersHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AddictionJournal of


Depression Research and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geography Journal


Research and TreatmentAutism


Economics Research International


The reminder of this paper is organized as followsSection 2 reviews major economic growth indexes andincome distribution indexes in the existing literatureSection 3 defines the concept of real national income averagegrowth rate and Section 4 further discusses the advantages ofusing this new index In Section 5 using the data in Taiwanwe compare this new index with the usual real economicgrowth rate Finally Section 6 concludes this paperwith someremarks

2 Economic Index

21 Measuring Economic Growth Index The main economicindex for measuring economic growth is real economicgrowth rate The formula of the real economic growth rateis defined as follows

( Real GDP119905

Real GDP119905minus1

) minus 1 (1)

Real Economic Growth Rate119905represents real eco-

nomic growth rate of 119905th yearReal GDP

119905represents real gross domestic product of

119905th yearReal GDP

119905minus1represents real gross domestic product

of 119905 minus 1th year

Gross domestic product can be explained as the powerof the country Positive real economic growth rate means thepower of the country is increasing It seems every countryaims to promote their real economic growth rate

Real economic growth rate is calculated as the totalincome of all nationals in the last year divided by the totalnet increase income of all nationals in this year It can revealthe macroeconomic level of the entire nation but it does notdisplay the information of the income growth rate of eachindividual nationHowever promoting real economic growthrate usually leads to a side effect unequal income distribution[1 2]

22 Measuring Income Distribution Index The main eco-nomic index for measuring income distribution includesLorenz curve Gini coefficient and ratio of income share ofhighest 20 to that of lowest 20

Lorenz curve is an income aggregation curve whichaggregates income from the lowest income national to thehighest income national [3]

The Gini coefficient is a measuring index of statisticaldispersion formulated by Corrado Gini [4] Gini coefficientis usually defined mathematically based on the Lorenz curveGini coefficient ranges from 0 to 1 A low Gini coefficientindicates a more equal distribution Gini coefficient = 0 rep-resents complete equality Higher Gini coefficients indicatemore unequal distribution Gini coefficient = 1 representscomplete inequality Although Gini coefficient can explainthe unequal degree of income distribution it is useless to useGini coefficient to express the financial net increase incomedistribute of each national in each year

As implied by the name ratio of income share of highest20 to that of lowest 20means the total income of nationalswhose income rank is in lowest 20 divides the total incomeof nationals whose income rank is in highest 20 [5] Thedrawback of ratio of income share of highest 20 to that oflowest 20 is that it does not consider the incomedistributionof nationals whose income is in the middle

3 Real National Income Average Growth Rate

To consider economic growth and incomedistribution simul-taneous this research provides a new indexmdashreal nationalincome average growth rateThe concept of RNIAGR is aver-aged real income growth of each individual national Unlikereal economic growth rate that calculates the economicgrowth rate based on the total income of all nationals inprevious year RNIAGR is calculated from individual incomegrowth rate based on individual income in the previous yearand average of all of individual income growth rate RNIAGRcan reflect the average income growth status of individualnational This formula supposes that the volume of nationalsin 119905th year and in 119905minus1th year in the sameCountry is the same

RNIAGR119905= 1119901 lowast119901

sum119895=1

( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1 (2)

RNIAGR119905represents real national income average

growth rate in 119905th year119868119895119905represents the income of 119895th national in 119905th year

119868119895119905minus1

represents the income of 119895th national in 119905 minus 1thyear119865119905represents the inflation rate in 119905th year

119875 represents the volume of nationals in the country

Using Figure 1 as example 119909-axis represents the rank ofincome from low income national to high income nationalPoint 119860 represents the lowest income national and point 119861represents the highest incomenational Frompoint119860 to point119861 all of the points can reflect an income of one national

If Area119860119861119862 represents the total income of all nationals inlast year Area 119860119862119863 represents the net increase income of allnationals in this year So nominal economic growth rate canbe represented as Area 119860119861119862 divides Area 119860119862119863 From point119860 to point 119861 all of the points can reflect an income of onenational Every national uses the individual income as a baseto calculate the individual income growth rate And we canobtain nominal national income average growth by averagingthe individual income growth rate For example point 119894 thatis between point 119860 and point 119861 represents 119894th national Theincome of 119894th national in last year is line 119894119895 The net increaseincome of 119894th national in this year is line 119895119896 The individualincome growth rate of 119894th national is line 119894119895 divide line 119895119896In order to reflect the real income status of each nationalnominal national income average growth needs to be dividedby inflation rate of this year to acquire real national incomeaverage growth rate of this year


A B

C

D

i

j

k

Citizen

Income

x-axis



FSRNIAGR119905

= (5

sum119894=1


119868119895119905


119868119895119905minus1

)


(3)

FSRNIAGR119905


119905th year119868119895119905minus1


A B

C

D

s1 s2 s3 s4

c1c2

c3

c4

d1

d2

d3

d4

Citizen

Income

x-axis




in 119905th year119901119905minus1







119905must be







1 11990411199042





















1119888111988821199042divides Area



FSRNIAGR119905

= ((119860 (11986011990411198891)119860 (11986011990411198881) +

119860 (1199041119889111988921199042)

119860 (1199041119888111988821199042) + 119860 (119904

2119889211988931199043)

119860 (1199042119888211988831199043)

+119860 (1199043119889311988941199044)119860 (1199043119888311988841199044) + 119860 (119904

41198894119863119861)

119860 (11990441198884119862119861) ) lowast (119875

119905minus1119875119905)

(1 + 119865119905) )













25000

20000

15000

10000

5000

0Low

nationalincome

High nationalincome

Low nationalincome

High nationalincome

Low nationalincome

High nationalincome















Increase




down more

Decrease



promote a few






119894=1119880(119868119894119905) where (119868

1119905 1198682119905 119868

119901119905) is the


119905minus1= (1198681119905minus1

1198682119905minus1

119868119901119905minus1


119901119905) and 1198681015840

119905= (11986810158401119905 11986810158402119905 1198681015840

119901119905) at time 119905 We


119894119905minus119868119894119905minus1

≪ 119868119894119905minus1

and 1198681015840119894119905minus

119868119894119905minus1

≪ 119868119894119905minus1

for 119894 = 1 2 119901 Let Δ119880119895119905equiv 119880(119868

119895119905) minus 119880(119868

119895119905minus1)

andΔ1198801015840119895119905equiv 119880(1198681015840

119895119905)minus119880(119868



sum119880(119868119895119905) minussum119880(1198681015840

119895119905)

= sum[119880 (119868119895119905minus1

) + Δ119880119894119905] minussum[119880 (119868

119895119905minus1) + Δ1015840

119895119905]

= sumΔ119880119895119905minussumΔ1198801015840

119895119905

(5)


119895119905asymp sum(120597120597119868)119880(119868

119895119905minus1) lowast 119889119868

119895and sumΔ1198801015840

119895119905asymp

sum(120597120597119868)119880(119868119895119905minus1

) lowast 1198891198681015840119895 where 119889119868

119895= 119868119895119905

minus 119868119895119905minus1

and

1198891198681015840119895

= 1198681015840119895119905

minus 119868119895119905minus1


119895119905) minus sum119880(1198681015840

119895119905) to be

[sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886

equiv sum 120597120597119868119880 (119868


120597119868119880 (119868119895119905minus1

) 1198891198681015840119895

= sum 119889119868119895minus 1198891198681015840119895

119868119895119905minus1

(6)




119905)


119895119905)]119886 ge 0

(7)



119905)

= 1119875 sum( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

minus 1119875 sum( 1198681015840

119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

= 1119875 (1 + 119865

119905) (sum

119868119895119905

119868119895119905minus1

minussum 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119868119895119905minus 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

(119868119895119905minus 119868119895119905minus1


)119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119889119868119895minus 1198891198681015840119895

119868119895119905minus1

)

(8)



119905) ge 0 lt==gt [sum119880(119868

119895119905) minus

sum119880(1198681015840119895119905)]119886 ge 0


6 Examples




Year






rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween


40


highest 20










800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal






A B

C

D

i

j

k

Citizen

Income

x-axis



FSRNIAGR119905

= (5

sum119894=1


119868119895119905


119868119895119905minus1

)


(3)

FSRNIAGR119905


119905th year119868119895119905minus1


A B

C

D

s1 s2 s3 s4

c1c2

c3

c4

d1

d2

d3

d4

Citizen

Income

x-axis




in 119905th year119901119905minus1







119905must be







1 11990411199042





















1119888111988821199042divides Area



FSRNIAGR119905

= ((119860 (11986011990411198891)119860 (11986011990411198881) +

119860 (1199041119889111988921199042)

119860 (1199041119888111988821199042) + 119860 (119904

2119889211988931199043)

119860 (1199042119888211988831199043)

+119860 (1199043119889311988941199044)119860 (1199043119888311988841199044) + 119860 (119904

41198894119863119861)

119860 (11990441198884119862119861) ) lowast (119875

119905minus1119875119905)

(1 + 119865119905) )













25000

20000

15000

10000

5000

0Low

nationalincome

High nationalincome

Low nationalincome

High nationalincome

Low nationalincome

High nationalincome















Increase




down more

Decrease



promote a few






119894=1119880(119868119894119905) where (119868

1119905 1198682119905 119868

119901119905) is the


119905minus1= (1198681119905minus1

1198682119905minus1

119868119901119905minus1


119901119905) and 1198681015840

119905= (11986810158401119905 11986810158402119905 1198681015840

119901119905) at time 119905 We


119894119905minus119868119894119905minus1

≪ 119868119894119905minus1

and 1198681015840119894119905minus

119868119894119905minus1

≪ 119868119894119905minus1

for 119894 = 1 2 119901 Let Δ119880119895119905equiv 119880(119868

119895119905) minus 119880(119868

119895119905minus1)

andΔ1198801015840119895119905equiv 119880(1198681015840

119895119905)minus119880(119868



sum119880(119868119895119905) minussum119880(1198681015840

119895119905)

= sum[119880 (119868119895119905minus1

) + Δ119880119894119905] minussum[119880 (119868

119895119905minus1) + Δ1015840

119895119905]

= sumΔ119880119895119905minussumΔ1198801015840

119895119905

(5)


119895119905asymp sum(120597120597119868)119880(119868

119895119905minus1) lowast 119889119868

119895and sumΔ1198801015840

119895119905asymp

sum(120597120597119868)119880(119868119895119905minus1

) lowast 1198891198681015840119895 where 119889119868

119895= 119868119895119905

minus 119868119895119905minus1

and

1198891198681015840119895

= 1198681015840119895119905

minus 119868119895119905minus1


119895119905) minus sum119880(1198681015840

119895119905) to be

[sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886

equiv sum 120597120597119868119880 (119868


120597119868119880 (119868119895119905minus1

) 1198891198681015840119895

= sum 119889119868119895minus 1198891198681015840119895

119868119895119905minus1

(6)




119905)


119895119905)]119886 ge 0

(7)



119905)

= 1119875 sum( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

minus 1119875 sum( 1198681015840

119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

= 1119875 (1 + 119865

119905) (sum

119868119895119905

119868119895119905minus1

minussum 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119868119895119905minus 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

(119868119895119905minus 119868119895119905minus1


)119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119889119868119895minus 1198891198681015840119895

119868119895119905minus1

)

(8)



119905) ge 0 lt==gt [sum119880(119868

119895119905) minus

sum119880(1198681015840119895119905)]119886 ge 0


6 Examples




Year






rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween


40


highest 20










800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal















1119888111988821199042divides Area



FSRNIAGR119905

= ((119860 (11986011990411198891)119860 (11986011990411198881) +

119860 (1199041119889111988921199042)

119860 (1199041119888111988821199042) + 119860 (119904

2119889211988931199043)

119860 (1199042119888211988831199043)

+119860 (1199043119889311988941199044)119860 (1199043119888311988841199044) + 119860 (119904

41198894119863119861)

119860 (11990441198884119862119861) ) lowast (119875

119905minus1119875119905)

(1 + 119865119905) )













25000

20000

15000

10000

5000

0Low

nationalincome

High nationalincome

Low nationalincome

High nationalincome

Low nationalincome

High nationalincome















Increase




down more

Decrease



promote a few






119894=1119880(119868119894119905) where (119868

1119905 1198682119905 119868

119901119905) is the


119905minus1= (1198681119905minus1

1198682119905minus1

119868119901119905minus1


119901119905) and 1198681015840

119905= (11986810158401119905 11986810158402119905 1198681015840

119901119905) at time 119905 We


119894119905minus119868119894119905minus1

≪ 119868119894119905minus1

and 1198681015840119894119905minus

119868119894119905minus1

≪ 119868119894119905minus1

for 119894 = 1 2 119901 Let Δ119880119895119905equiv 119880(119868

119895119905) minus 119880(119868

119895119905minus1)

andΔ1198801015840119895119905equiv 119880(1198681015840

119895119905)minus119880(119868



sum119880(119868119895119905) minussum119880(1198681015840

119895119905)

= sum[119880 (119868119895119905minus1

) + Δ119880119894119905] minussum[119880 (119868

119895119905minus1) + Δ1015840

119895119905]

= sumΔ119880119895119905minussumΔ1198801015840

119895119905

(5)


119895119905asymp sum(120597120597119868)119880(119868

119895119905minus1) lowast 119889119868

119895and sumΔ1198801015840

119895119905asymp

sum(120597120597119868)119880(119868119895119905minus1

) lowast 1198891198681015840119895 where 119889119868

119895= 119868119895119905

minus 119868119895119905minus1

and

1198891198681015840119895

= 1198681015840119895119905

minus 119868119895119905minus1


119895119905) minus sum119880(1198681015840

119895119905) to be

[sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886

equiv sum 120597120597119868119880 (119868


120597119868119880 (119868119895119905minus1

) 1198891198681015840119895

= sum 119889119868119895minus 1198891198681015840119895

119868119895119905minus1

(6)




119905)


119895119905)]119886 ge 0

(7)



119905)

= 1119875 sum( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

minus 1119875 sum( 1198681015840

119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

= 1119875 (1 + 119865

119905) (sum

119868119895119905

119868119895119905minus1

minussum 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119868119895119905minus 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

(119868119895119905minus 119868119895119905minus1


)119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119889119868119895minus 1198891198681015840119895

119868119895119905minus1

)

(8)



119905) ge 0 lt==gt [sum119880(119868

119895119905) minus

sum119880(1198681015840119895119905)]119886 ge 0


6 Examples




Year






rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween


40


highest 20










800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal









Increase




down more

Decrease



promote a few






119894=1119880(119868119894119905) where (119868

1119905 1198682119905 119868

119901119905) is the


119905minus1= (1198681119905minus1

1198682119905minus1

119868119901119905minus1


119901119905) and 1198681015840

119905= (11986810158401119905 11986810158402119905 1198681015840

119901119905) at time 119905 We


119894119905minus119868119894119905minus1

≪ 119868119894119905minus1

and 1198681015840119894119905minus

119868119894119905minus1

≪ 119868119894119905minus1

for 119894 = 1 2 119901 Let Δ119880119895119905equiv 119880(119868

119895119905) minus 119880(119868

119895119905minus1)

andΔ1198801015840119895119905equiv 119880(1198681015840

119895119905)minus119880(119868



sum119880(119868119895119905) minussum119880(1198681015840

119895119905)

= sum[119880 (119868119895119905minus1

) + Δ119880119894119905] minussum[119880 (119868

119895119905minus1) + Δ1015840

119895119905]

= sumΔ119880119895119905minussumΔ1198801015840

119895119905

(5)


119895119905asymp sum(120597120597119868)119880(119868

119895119905minus1) lowast 119889119868

119895and sumΔ1198801015840

119895119905asymp

sum(120597120597119868)119880(119868119895119905minus1

) lowast 1198891198681015840119895 where 119889119868

119895= 119868119895119905

minus 119868119895119905minus1

and

1198891198681015840119895

= 1198681015840119895119905

minus 119868119895119905minus1


119895119905) minus sum119880(1198681015840

119895119905) to be

[sum119880(119868119895119905) minussum119880(1198681015840

119895119905)]119886

equiv sum 120597120597119868119880 (119868


120597119868119880 (119868119895119905minus1

) 1198891198681015840119895

= sum 119889119868119895minus 1198891198681015840119895

119868119895119905minus1

(6)




119905)


119895119905)]119886 ge 0

(7)



119905)

= 1119875 sum( 119868119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

minus 1119875 sum( 1198681015840

119895119905

119868119895119905minus1

lowast (1 + 119865119905)) minus 1

= 1119875 (1 + 119865

119905) (sum

119868119895119905

119868119895119905minus1

minussum 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119868119895119905minus 1198681015840119895119905

119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

(119868119895119905minus 119868119895119905minus1


)119868119895119905minus1

)

= 1119875 (1 + 119865

119905) (sum

119889119868119895minus 1198891198681015840119895

119868119895119905minus1

)

(8)



119905) ge 0 lt==gt [sum119880(119868

119895119905) minus

sum119880(1198681015840119895119905)]119886 ge 0


6 Examples




Year






rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween


40


highest 20










800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal







Year






rank isbetween

lowest 20and lowest

40


rank isbetween

lowest 40and lowest

60


rank isbetween


40


highest 20










800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal






800

600

400

200

000

minus200

minus400

minus600

minus800

minus1000

1995

1996

1997

1998

1999

2000

2001

2002

2003

2005

2004

2006

2007

2008

2009

2010

2011





()










References















Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal













Psychiatry Journal



















Volume 2014


AddictionJournal of




Geography Journal





research article real national income average growth rate

Documents