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Tuesday, September 06, 2005

Educational Equity in Peru (& Colombia)

Economic Value of Education

Course Assignment 2

Select a country that you wish to study for this assignment.
1. Set out the criteria that you would advocate should be used to assess the equity of the educational system of your country and justify your selection of criteria.
2. Collect and present data on some of the equity criteria you have selected in Part 1. Assess what the evidence indicates about the equity of the education system and how adequate this evidence is.

Country Selected: Peru (& Colombia)

Weijie Ng
MA Economics of Education

“Equity is a roguish thing. For Law we have a measure, know what to trust to; Equity is according to the conscience of him that is Chancellor, and as that is larger or narrower, so is Equity.”
(Selden, J 1847)

In Section A, equity, with distinctions made between different conceptions of equity, will first be broadly discussed. Subsequently, several types of dispersal measures will be introduced in Section B and, it will be argued that among them, the Gini coefficient is most appropriate as a measure of educational equity. Then, in Section C, the Gini coefficient will be used to analyse the equity of the Peruvian school system. It will also be used to calculate equity in a comparator nation, Colombia, in order to give a level of perspective to the Peruvian results. Section D will sum up the results of the analyses, and outline how equity might still be inadequately measured.

Section A: The meaning of equity

Equity, in economics, has two broad meanings. One refers to the capital of firms while the other, of which we are concerned with in this paper, is related to fairness and justice: fairness in dividing the economic pie, justice in accruing both benefits and costs to individuals, etc.

A distinction between horizontal and vertical equity has been made (Musgrave 1959). Horizontal equity means that equals should be treated equally. For example, in education, if all students, regardless of socio-economic background, gender, etc. are deemed to be equal, then horizontal equity might require identical treatment of all of them. Vertical equity means that it may be fair for ‘unequals’ to be treated unequally. Monk (1990) suggested that the division, in modern education, of students into academic and vocational pathways, on the basis of differing needs and aptitudes, can be viewed as one example of vertical equity. Both the equal treatment of equals and the unequal treatment of ‘unequals’ may rightfully be fair, but not the unequal treatment of equals nor the equal treatment of unequals (Musgrave 1959). Avoiding the latter two or indeed, the identification of inequity, is unfortunately hardly simple. A major problem lies in the definitions of ‘equals’ and ‘unequals’. In Monk’s (1990) example, students with different needs and ability are deemed to be unequal, and therefore, differential and unequal treatment is justifiable. However, others may deem that ALL students, regardless of any background factors such as need and ability, are equal. Accordingly, unequal treatment of equal students would neither be horizontally nor vertically equitable.

There is also a difference between ideas of procedural and distributional equity. Procedural equity[1] may be defined as the correct and uniform adherence to and application of rules in such a way that shows consistency and even-handedness (Barry, 1990). These rules may be formal or informal, explicit or implicit, and the processes involved in their application may be covert or overt. Procedural equity is linked to ‘equality of educational opportunity’, which ‘exists when a child’s opportunity does not depend upon either his parents’ economic circumstances or his location within the state’ (Wise, 1967). The processes of cream-skimming for more able pupils through the clandestine use of religious interviews (West & Hind, 2003) at the expense of the ‘less able’ working class, and the institutional racism inherent in ‘tiering’ which discriminate against black students (Gillborn 2001) reduces the opportunities available to some groups relative to others are examples of procedural inequity. On the other hand, distributional equity refers to the distribution of both inputs / costs, such as teacher-hours (labour input), expenditure, etc. and outputs / benefits, which in education would include pecuniary benefits from education as well as external benefits (Wolfe, 1995) that are difficult to measure or accrue to education.

Section B: Measuring equity: the methods

Given that there are problems with the identification of equals and ‘unequals’, that inequitable practices may be covert, and that measurement and accrual of expenditures on and benefits of education are difficult, it will perhaps be of no surprise that the measurement of equity is hardly a simple thing. There are plenty of measures that have been used to analyse educational equity, ranging from Gini coefficients to bar chart comparisons and interquartile ranges to desegregation indices, each with a different method of computation and set of ‘pros’ and ‘cons’. When different measures are used and different computations are made, one may expect to arrive at different verdicts. For example, different studies of the English school system after the Education Reform Act of 1988 for example have given different pictures: Gorard & Fitz’s (1998) desegregation index suggested an overall improvement in equity while Noden’s (2000) isolation index showed a consistent decline over the same period.

In this section, a number of equity measures will be critically explored, with the understanding that no one single measure (or perhaps even set of measures) is sufficient in itself as a comprehensive measure of equity. However, the best among them may still serve as an indicator and a rough guide. Later, this indicator will be used in the following Section C to analyse equity in the Peruvian school system.

Where only univariate data, i.e. data for only a single variable such as national test scores, is available, a few statistical measures are useable as an indicator of dispersion and inequality. Range is one of them, and is perhaps the simplest: it involves merely the subtraction of the smallest value from the biggest value. The larger the range, the larger is the disparity and therefore the larger is the inequality. Its simplicity can be a boon because it is very quick and easy to compute. However, since only the two extreme values are taken into account, the range as a measure of dispersion may be skewed easily. Moreover, the range gives little information about the variability between the two extremes.

A more useful measure for univariate date would be the inter-quartile range, which has been used in the OECD Education At A Glance 2004 report. Essentially, to find the inter-quartile range of a sample group, the individuals in the sample have to be first ranked in ascending (or descending) order and divided into four equal groups, each called a quartile. The group with the lowest values is the first quartile, while that of the highest group is the fourth quartile. Then:

Equation here

Just like the range, the larger the inter-quartile range, the larger is the disparity and therefore the larger is the inequality. Because the inter-quartile range does not take into account of the highest and lowest values in the whole sample in its computation, it is less easily influenced by outliers and is therefore arguably more stable and credible as a measure of inequality. Yet, it still does not take into account the variability of values in the middle nor at the extremes (Blalock, 1979).

Yet another measure that can be used for univariate data, and has also been used in Education At A Glance 2003 (OECD, 2003) is that of variance. Variance is ‘a measure of spread in the distribution of a random variable’ (Wooldridge, 2003), and a larger variance indicates less equality. Mathematically, where n refers to the number of samples observed, variance may be expressed as the following:

Equation here

Its core advantage over the first two measures is that it takes into account every single value in the whole sample. Unfortunately, variance is not only influenced by differences in the sample values, but also the scale of the values, which violates the criterion of scale invariance, one of the four criteria that James & Taubber (1985) argued to be required of a good measure of inequality and segregation. For example, a reconfiguration of the unit of measurement that doubles the numerical value of every sample quadruples the variance. Also, variance is still influenced by the presence of outlying values.

When data is richer, and includes information on individual background socio-economic characteristics, multi-group and multivariate measures may be applied. Amongst them is the popular Gini co-efficient[2], which has been used extensively to examine inequalities in income, but not quite so much in education (Thomas et al. 2000). The educational Gini coefficient may be understood using Figure 1. The Lorenz curve shows, for the bottom x% of households, the percentage y% of the educational rewards which they have. The percentage of households is plotted on the x-axis, the percentage of educational rewards on the y-axis. The line of equality shows the position of the Lorenz curve when each household is allocated an equal quantity of educational rewards. Then, the Gini coefficient is the area between the line of perfect equality and the Lorenz curve (Area A), as a percentage of the area under the line of perfect equality (Areas A+B). If the Gini coeefficient is zero, then educational rewards are distributed equally. As the Gini coefficient increases, there is greater inequality. If equality is deemed to be equitable, then there is greater inequity as well.

Figure 1: Illustration of Lorenz curve here

There are other methods of calculating the Gini coefficient, and Thomas et al. (2000) have devised algebraic formula such as the following:

Equation here

(Thomas et al. 2000, page 9)

The Gini coefficient has an edge over other measures of equality, such as variance and relative mean deviation, because it is scale invariant (James & Taubber, 1985) and fulfils Dalton’s (1920) principle of transfers. This principle suggests that measures of inequality should increase whenever educational rewards are transferred from a relatively deprived household to another wealthier one, and therefore has substantial intuitive appeal. However, the Gini coefficient is not without problems. For example, when Lorenz curves cross, the Gini coefficient gives ambiguous results that cannot be easily compared. Also, as Barr (2004) would argue, the Gini coefficient is really a weighted sum of household’s educational rewards, with the weights determined solely, & rather arbitrarily, to be the household’s rank order in the distribution.

There are many other measures of inequality, dispersion and segregation that may be used to analyse educational equity, such as the Theil index (1972), desegregation index (Gorard, 1998), etc. Choice between different measures of inequality can make a difference, even when using the same data. Atkinson (1970) showed that rank ordering of countries by income inequality can differ substantially using different inequality measures. This is not only because the choice between the different measures is a choice between alternative methods of measuring the same thing, but also can be, as Allison (1978) argued, ‘a choice among alternative definitions of inequality’.

It is not the remit of this report to comprehensively detail all available measures of equity, but to set out criteria for the assessment of a national educational system’s equity. Thus far, several measures have been discussed, of which arguably the Gini coefficient has emerged to be the most robust methodologically and is yet still reasonably simple to calculate. Even though it does have its cons, the Gini coefficient may still give a good first approximation of educational equity. Therefore, this will be the measure with which to attain a preliminary analysis of the fairness of the Peruvian education system in Section C.

Section C: Educational equity in Peru (& Colombia)

For the purposes of measuring and comparing equity in the school systems of Peru (and Colombia), Edstats, the database developed and maintained by the Education Group of the Human Development Network (HDNED) of the World Bank[3], has been utilised. Equivalent measurements for Colombia will be in brackets next to those for Peru.

Assuming that each year of formal schooling is worth the same, no matter how old the recipients were when they received the education, or the level of formal schooling, etc., and using a notional formal school years (FSH) as a unit of educational rewards, the following table and Lorenz curve may be derived for adults aged 15-24 in Peru in 1996:

Table 1: Distribution of FSH in relation to households by income [Peru, 1996, 15-24 year-olds] here

Figure 2: Lorenz curve for distribution of FSH in relation to households by income, [Peru, 1996, 15 – 24 year olds] here

To calculate the Gini coefficient, G for this distribution, perhaps the easiest way is to use Figure 2 as a starting point, and calculate the area of the various trapezoids in the diagram:

Area under the line of equality:
½ X Base X Height: ½ X 100 X 100 = 5000

Area under the Lorenz curve:
½ X (Sum of Heights of Triangle & Trapezoids) X Base
= ½ X [100 + 2 (12.72 + 30.28 + 51.65 +75.06)] X 20
= ½ X 439.44 X 20
= 4394.4

G = Area A / (Area A + Area B)
= (5000 – 4394.4) / 5000
= 0.121

From the above calculations, the Gini coefficient for the distribution of formal schooling hours in relation to households by income, for Peruvian (Colombian) 15 – 24 year olds in 1996 is: 0.121 (0.158).

Using similar calculations, the corresponding Gini coefficient for Peruvian (Colombian) above 25 year olds in the same year is: 0.231 (0.248).

Given that primary schooling has been shown to have both higher social and private rates of return to education (Psacharopoulos, 1985), suppose that the value of primary formal schooling hours[4] are adjusted to be 1.5[5] times of any further education thereafter, the distribution of adjusted formal schooling hours will be as follows:

Table 2: Distribution of adjusted FSH in relation to households by income [Peru, 1996, 15–24 year-olds] here

The corresponding Gini coefficient for the distribution of adjusted formal schooling hours in relation to households by income, for Peruvian (Colombian) 15 – 24 year olds in 1996 is: 0.096 (0.128).

Using similar calculations, the corresponding Gini coefficient for adjusted Peruvian (Colombian) above 25 year olds in the same year is: 0.202 (0.220).

Section D: Summing up: Where is there more equity? Really?

From the results in Section C, the Gini coefficients are all closer to 0 than to 1, and one might therefore postulate that the distribution of educational resources is more equal than unequal in both Peru and Colombia. Colombian G values have also been calculated for comparison purposes, and since G is lower in Peru than in Colombia, this implies that the Peruvian school system is more equal across socio-economic class than that of the Colombia, and arguably more equitable. Even when formal schooling hours were re-weighted to take into account a probable higher benefit to primary schooling than post-primary schooling, similar results and conclusions were derived.

Going back to the discussion on equity in Section A, the Gini coefficient is an indicator of horizontal equity, when society deems that it is fair and just for everyone, regardless of family income, to undergo the same number of years of formal schooling, but it actually says little about vertical equity. G may be used to measure whether equals are treated equally but not whether unequals are treated unequally. Also, clearly, the Gini coefficient can only be used to examine what can be observed and evaluate distributional equity, but not procedural equity, which can be both covert and overt, and is moreover difficult to quantify. To examine procedural equity, there is a strong case for micro level sociological, ethnographic work, which relies primarily on exhaustive study of individual cases. The core benefit of case studies is that contextual information is mined, which allows for greater understanding of causal processes (De Vaus, 2001).

More reservations need to be made about the G findings above. Firstly, as Behrman and Birdsall (1983) argue, quantity alone is not enough and quality must be taken into consideration. G probably understates inequalities in both Peru and Colombia, as it is to be expected that the quality of schooling for the richer would be better than that for the poorer. Also, thus far, the trustworthiness of the data has been implicitly relied upon. If politicians and bureaucrats are capable of massaging unemployment figures, why not also educational data?

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Atkinson, AB (1970). ‘On the measurement of inequality’, Journal of Economic Theory, Vol. 2, 244-263

Barr, N (2004). Economics of the welfare state 4th edition, Oxford: Oxford University Press.

Barro, R and Lee, JW (1996). ‘International measures of schooling years and schooling quality’, American Economic Review, Vol. 86 (2), 218-223

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Behrman, J and Birdsall, N (1983). ‘The quality of education. Quantity alone is misleading.’, American Economic Review, Vol. 73, No. 5, 928-946.

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OECD (2004). Education at a glance 2004, Paris: OECD

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Wise, AE (1967). Rich schools poor schools: the promise of equal educational opportunity, Chicago: University of Chicago Press.

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[1]‘Procedural fairness’ in Barry, (1990)
[2] The Gini coefficient may be and has also been used for uni-variate data, such as that of incomes.
[3] Available online at
[4] 6 primary school years in Peru (and Colombia).
[5] Arbitrarily chosen


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