DOCUMENT RESUME

ED 103 979 EA 006 903

AUTHOR Garas, Walter I.TITLE Use of the Lorenz Curve and Gini Index in School

Finance Research.PUB DATE Mar 75NOTE 19p.; Paper presented at the Annual Meeting of the

American Educational Association (60th, Washington,D.C., March 31-April 4, 1975)

EDRS PRICE MF-$0.76 HC-$1.58 PLUS POSTAGEDESCRIPTORS Assessed Valuation; *Educational Finance; Elementary

Secondary Education; *Expenditure Per Student;Expenditures; *Finance keform; Fiscal Capacity;Income; *Statistical Studies; Statistics; Tax Effort;Tax Rr.tes

IDENTIFIERS Gini Coefficient; Lorenz Curve

ABSTRACTThis paper proposes two modifications of the Lorenz

technique for measuring inequalities in financing education. One isto order expenditures (and tax rates) by district fiscal abilitybefore calculating the cumulative percentages. It is possible, as aresult, to get a Lorenz curve that crosses the diagonal line, whichgives some useful insights. The other is to calculate Ginicoefficients for both expenditures and tax rates and to consider thesus of the two coefficients to be a measure of the total inequalityin the system. (Author)

a

USE CF THE LORENZ CURVE AND GINI INDEX IN SCHOOL FINANCE RESEARCH

DE PuTNIE NT OF HIALTN.EDUCATION TEELFARENATIONAL INS MUTE t:

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Walter I. Garms'

University of Rochester

March, 1975

Cr" (Prepared for presentation at the 1975 Annual Meeting of the American

cy. Educational Research Association, Washington, D.C., March 30-April 3, 1975)

C)r.4

C.73 Those interested in school finance have long been interested in

LAI inequalities of educational expenditure and educational effort. Economists

have long been interested in inequalities of income. It is unfortunate

that there has not been more communication between these two groups of

people, for the economists have one standard technique for looking at

income inequalities that can be very useful in examining inequalities in

educational expenditure and effort. That technique is the use of the Lorenz

curve and its associated Gini coefficient. However, the use of the Lorenz

curve in the way it is usually used by economists would give misleading

results as a measure of inequalities in school finance. It is the purpose

of this paper to describe and illustrate a modification of the Lorenz

technique that yields some unique insights for those interested in school

finance research.

As used in the measurement of income inequality, the Lorenz technique

is to array all individuals in order of ascending income. Then at convenient

points (for each one percent of the total individuals, say) the cumulative

percent of individuals and the cumulative percent of total income is

calculated. As an example of how this is done, the information for families

1

2

in the United States in 1969 is given in Table I. The cumulative

percentages are then plotted on a graph called a Lorenz chart. The

horizontal axis is the cumulative percentage of families, running from 0

to 100, and the vertical axis is the cumulative percentage of income, also

running from 0 to 100. The data given above have been plotted on a Lorenz

chart (Figure 1), and are represented by the curved line. The straight

diagonal line represents complete equality of incomes. That is, if all

families received the same income, the "lowest" ten percent of the families

would receive ten percent of the income, the "lowest" fifty percent would

receive fifty percent of the income, and so on. The extent to which the

curved line sags below the diagonal line, then is a measure of the

inequality of incomes. This measure is formalized in the Gini coefficient,

which is the ratio of the area bounded by the diagonal line and the curved

line, divided by the area bounded by the diagonal line, the horizontal axis,

and the right-hand vertical line. A Gini coefficient of zero would represent

absolute equality,and a Gini coefficient of 1.00 would represent absolute

inequality (the situation in which one individual got all of the income and

no one else got any). The Gini coefficient for the family income data

displayed above is about 0.36.

There are, of course, other measures of inequality that can be used.

They .1clude the range of incomes (the highest less the lowest), the

interquartile range (the difference between income at the third quartile and

the first quartile), the interquartile range divided by the mean, the standard

deviation, and the coefficient of variation (the standard deviation divided

by the mean). Each is a better measure than the previous one, at the cost

of increasing difficulty of computation and of explaining to a lay audience.

The range takes into account only two incomes, the lowest and the highest.

Table I

Income

Rank

Family Income in the U.S., 1969

Percent of Cumulative Percent of

Total Income Families Income

Lowest tenth 1 10 1

Second tenth 3 20 4

Third tenth 5 30 9

Fourth tenth 6 40 15

Fifth tenth 8 50 23

Sixth tenth 9 60 32

Seventh tenth 11 70 43

Eighth tenth 12 80 55

Ninth tenth 16 90 71

Highest tenth 29 100 100

100

90

80

70

Percent6o

of 50

Income

40

30

20

10

Figure I

10 20 30 40 50 60 70 80 90 100

4sr

Percent of Families

3

4

Thus this most widely used measure of inequal:ty is only a measure of two

extremes that are almost certain to be unusual cases. The interquartile

range ignores these extremes, thus giving a more stable measure at the

expense of ignoring half of the data. The interquartile range divided by

the mean is simply a way of standardizing the interquartile range so that

distributions with different means may be validly compared. The standard

deviation (and its standardized version, the coefficient of variation)

takes into account all of the data. Its principal drawback is that

condensing all of the data into a single statistic does not allow us to

look at any other characteristics of the distribution. The Lorenz curve

allows visual inspection of the characteristics of the distribution, and

the Gini index condenses the information on inequality into a single

statistic that is comparable across distributions with different means.

The difficulty with applying the Lorenz curve to expenditures per

student in the public schools is that not all of the differences in

expenditure are undesirable. At the same time that states are engaged in

equalization programs to reduce the differences in expenditure caused by

differences in fiscal ability, they are subsidizing programs for special

students that result in higher expenditures for them than for normal

students. Unfortunately, the results of these two kinds of subsidization

are combined in a single statistic called expenditures per student. Part

of the difficulty in trying to measure inequalities of educational

expenditure is the difficulty of separating expenditures for normal students

from expenditures for mentally retarded, physically handicapped, occupational

education, and for children in necessary small schools. Measures such as

the range, interquartile range, and standard deviation are incapable of doing

this. The Lorenz curve, as traditionally used by economists, is also

5

5

Table 2

DistrictNo. Students

Expendituresper Student

A.V. perStudent

Cumulative Percent ofStudents Expenditures

1 1000 800 20,000 10 72 2000 900 40,000 30 21

3 500 1000 60,000 35 25

500 1100 80,000 40 30

56

1000

1000

12001300

100,00090,000

5060

4050

78

2u00500

1400

1500

70,00050,000

8085

7380

9 500 1600 30,000 90 86

to 1000 1700 10,000 100 too

100

90

8o

70Percent

of 6o

Expen-diture 50

40

30

20

10

0

Figure 2

0 10 20 30 40 50 60 70

Percent of Students

80 90 100

Gini = .122

C

6

incapable of making this distinction. Table 2 and Figure 2 give fictitious

data for a sample of ten districts, and the Lorenz curve based on that data.

As with the chart of family incomes, the data have been arrayed in order

of increasing expenditure per student, and the cumulative percentage of

students and of expenditure is calculated. The data on assessed valuation

per student is included for information purposes, but is not used in the

calculations. Since there is no way of knowing how much of the expenditure

in a particular district is the result of differences in fiscal ability,

how much the result of differences in community desires for education, and

how much the result of differences in expenditures for special students,

the chart is interesting, but not especially informative.

If, on the other hand, we choose to array our districts in order of

increasing fiscal ability, the Lorenz chart can then give us information

about the extent to which differences in expenditures are correlated with

differences in fiscal ability. In Table 3 and Figure 3 this has been done.

The data on assessed valuation per student were purposely chosen to be as

uncorrelated as possible with expenditures per student. The result shows

in the Lorenz chart, where the Lorenz curve is much closer to the diagonal

line.

Figure 3 also shows a possible consequence of the use of this technique:

part of the Lorenz curve lies above the diagonal line. In the conventional

use of the Lorenz curve it is impossible for this to happen. But this is

a virtue of this variation of the technique. We can see at a glance, for

example, that the ten percent of the districts that are lowest in fiscal

ability actually spend 14 percent of the total money spent. It can readily

be seen that, with this technique, it is possible to have substantial

differences in expenditure, yet have the Lorenz curve lie close to the

7

Table 3

DistrictNo Students

Expendituresper Student

A.V. perStudent

Cumulative Percent ofStudents Expenditures

10 1000 170o 10,000 10 14

1 1000 800 20,000 20 20

9 500 1600 30,000 25 27

2 2(30 9co 40,000 49 42

8 500 1500 50,000 50 48

3 500 1000 60 ) 55 52

7 2000 1400 7C. dO 75 75

500 1100 80,000 80 80

6 1000 1300 90,000 90 90

5 1000 1200 100,000 100 100

100

90

80

70

Percent

of60

Expen-diture

50

40

30

20

10

0

Figure 3

0 10 20 30 40 50 60 70 80 90 -100

Gini . 0.004

Percent of Students

8

diagonal line. In calculating the Gini Index, it is appropriate to think

of areas between the Lorenz curve and the diagonal line that lie above the

line as negative areas. The result is that there could be substantial

deviation of the Lorenz curve from the diagonal line, but if this were the

result of differences in expenditure that were uncorrelated with differences

in fiscal ability, the Gini Index would be close to zero. Indeed, that is

so here, where the Gini Index is only 0.004, compared with an Index of

0.122 using the same data differently arrayed in Figure 2.

It should be particularly noted that the basic data used in Figures 2

and 3 are identical, and the axes of the Lorenz charts still have the same

labels. Only the method of ordering the data before calculating the

cumulative percentages is different.

We have been examining one side of the coin of fiscal neutrality, that

expenditures should not be a function of community wealth. The other side

is that tax effort should not be a function of community wealth. The Lorenz

curve can be used to examine this in an analogous fashion, although the

meaning of a cumulative percentage of tax rates is not as intuitively obvious.

In Table 4 and Figure 4, tax rates have been calculated for the districts in

our sample assuming a power equalizing system with full recapture that

guarantees an expenditure of $100 per student for each mill of tax rate.

It can be seen that the tax rates have a correlation of 1.00 with expenditures,

and thus the Lorenz curve for tax rates looks just like the Lorenz curve for

expenditures. However, it has an "opposite" meaning. That ;s, in the same

way that high expenditures per student and high family incomes are considered

to be "good", high tax rates are considered "bad." Thus, where an invidious

discrimination would tend to make the Lorenz curve of expenditures fall below

the diagonal, it will make the Lorenz curve of tax rates lie above the

9

9

Table 4

Power Equalized System

Dist. Studs.

Exp./Stud.

AV per Tax

Stud. Rate

Cumulative Percent ofStuds. Expends. Tax Rates

10 1000 1700 10,000 7 10 14 14

1 1000 800 20,000 8 20 20 20

9 500 1600 30,00C 6 25 27 27

2 2000 900 40,000 9 45 42 42

8 500 1500 50,000 5 50 48 48

3 500 1000 60,000 0 55 52 52

7 2000 1400 70,000 4 75 75 75

4 500 1100 80,005 1 80 80 80

6 1000 1300 90,000 3 90 90 90

5 1000 1200 100,000 2 100 100 100

Percent

of

Tax Rates

100

90

80

70

60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100

Figure 4

Percent of Students10

10

diagonal. In order to make the Gini coefficients comparable, then,

between exrcr.jitures and tax rates, the sign of the Gini coefficient for

tax rate should be reversed. The Dint coefficient for tax rates in this

case is then -0.004. If the two coefficients, for expenditures and tax

rate, are added together it should give a measure of the total inequality

in the system. In this case the sum of the two coefficients is exactly

zero, confirming the idea of a system that is completely neutral fiscally,

even though expenditures vary.

In Table 5 and Figure 5 the tax rates have been calculated assuming

instead that there is a state flat grant of $500 per student, with no

equalization. The tax rates have been calculated on the assumption that

all of the rest of the money must be raised by local taxes. Of course it

is highly unlikely, in such a situation, that District 10 would continue

to spend $1700 per student, taxing itself at 120 mills. But this has been

allowed to remain to illustrate the difference that the two approaches make

in tax rates when expenditures are not changed. The Gini coefficient for

tax rates in this case is 0.435, and the sum of the Gini coefficients for

expenditures and tax rate is 0.439.

(please see page 11)

11

Table 5

$500 Flat Grant

Dist. Studs.

Exp./Stud.

AV perStud.

TaxRate

Cumulative Percent ofStuds. Expends. Tax Rates

10 1000 1700 10,000 120 10 114 51

1 1000 800 20,000 15 20 20 58

9 500 1600 30,000 37 25 27 66

2 2000 900 40,000 10 45 42 74

8 500 1500 50,000 20 50 48 79

3 500 1000 60,000 8 55 52 80

7 2000 1400 70,000 13 75 75 91

4 500 1100 80,000 8 80 80 936 1000 1300 90,000 9 90 90 97

5 1000 1200 100,000 7 100 100 100

Figure 5

Percent 70

of60

Tax Rates

0 10 20 30 40 50 60 70 8o 90 100

Percent of Students JLZ!

12

It seems possible that the aggregate of decisions of the separate

school districts might be in a direction that would tend to minimize the

sum of the Gini coefficients for expenditures and tax rates. Continuing

with the assumption of a $500 flat grant per student, and using the same

type of table calculation as previously used, the following Gini

coefficients were obtained:

ExpendituresTaxRates

Expendituresplus Tax Rates

Assuming equal expendituresof $1200

(tax rates ranged from 7 to

0.000 0.373 0.373

70 mills)

Assuming equal tax rates of 0.158 0.000 0.15813 mills

(expenditures ranged from $630to $1800)

Assuming an in-between position(tax rates ranged from 10 to

0.104 0.127 0.231

20 mills, and expenditures from$700 to $1500)

The same thing was tried assuming that there was no state aid, and that

all income was raised through local taxes, with the following results:

ExpendituresTax

RatesExpenditures

plus Tax Rates

Assuming equal expenditures of 0.000 0.378 0.378$1200

(tax rates ranged from 12 to120 mills)

Assuming equal tax rates of 20mills

(expenditures ranged from $200to $2000)

0.267 0.000 0.267

Assuming an in-between position(tax rates ranged from 16 to 40

0.149 0.164 0.313

mills, and expenditures from$400 to $1600)

13

13

it appears from this very preliminary investigation that in an

unequalized system the sum of district decisions that results in least

variation in tax rates results in the smallest combined Gini coefficient.

In applying this technique, it is only appropriate to use assessed

valuation per student as an ordering device in states where there has been

almost complete equalization of assessment ratios statewide. Otherwise,

the appropriate measure to use is full value, or equalized value, per student;

the appropriate measure of tax effort, then, is tax rate on equalized value.

Some charts of expenditures per student and of tax rates for Oregon

districts are appended (Figures 7 to 9 ) as an illustration of the use of

the technique with real data. Assessments are carefully equalized in Oregon,

so that assessed valuation can confidently be used as a measure of fiscal

ability. One of the things that may immediately be noted is the nearness

of the Lorenz curves to the diagonal line, compared with the curve in

Figure I for family income in the United States. This is only partially a

result of the state equalization program in Oregon. It is also an artifact

of our method of measurement of educational expenditures which makes the

implicit assumption that all students in a particular school district have

exactly the same amount spent on them. The result of the aggregation is a

vast leveling of expenditure discrepancies. The analogous case is that of

constructing a Lorenz curve of income inequalities by region of the U.S.

If this is done for the four regions into which the country is divided by

the U.S. Statistical Abstract, with the assumption made that all families

within each region receive the same income, the results are as shown in

Table 6 and Figure 6. This Lorenz chart and Gini coefficient should be

compared with Figure I. Of course this aggregation error is not peculiar

to the Lorenz technique, but will affect aro/ measure of inequality one

1 4t

might choose to use.

Table 6

Family Income in the U.S., 1969, by Region

Median Cumulative Percent of

Families (000) Income Families Income

South 19,247 $ 8,105 30 26

Northeast 15,461 10,018 55 52

North Central 17,537 10,020 82 81

West 11,172 10,037 100 100

Percent

of

Income

100

90

80

70

60

50

40

30

20

10

Figure 6

0 10 20 30 40 50 60 70 80 90 100

Percent of Families

Gini = 0.038

14

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In summary, this paper proposes two modifications of the Lorenz

techn;que for measuring inequalities in financing education. One is

to order expenditures (and tax rates) by district fiscal ability before

calculating the cumulative percentages. It is possible, as a result,

to get a Lorenz curve that crosses the diagonal line, which gives

some useful insights. The other is to calculate Gini coefficients for

both expenditures and tax rates, and consider the sum of the two

coefficients to be a measure of the total inequality in the system.