Directory UMM :Data Elmu:jurnal:S:Socio-Economic Planning Sciences:Vol34.Issue1.Mar2000:

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Ecient provision of child quality of life in less developed

countries: conventional development indexes versus a

programming approach to development indexes

Raymond Raab

a,

*, Pradeep Kotamraju

b

, Stephen Haag

c a

Department of Economics, University of Minnesota-Duluth, Duluth, MN 55812, USA b_{Minnesota State Colleges and Universities System, St. Paul, MN 55101, USA}

c

Daniels College of Business, University of Denver, Denver, CO 80208, USA

Abstract

Using a linear programming approach, we establish a child quality of life (CQL) index by evaluating the ability of a less developed country (LDC) to maximize speci®c child development goals subject to minimizing speci®c resource availability indicators. This approach Ð which ranks LDCs from the most robustly ecient to the most robustly inecient in their ability to maximize goals while minimizing resource utilization Ð avoids using equal or subjective weights employed in conventional ranking schemes. The ranking of the 38 LDCs yields unexpected results and suggests a very dierent way of measuring and evaluating development policy.#2000 Elsevier Science Ltd. All rights reserved.

Keywords:DEA; Development indexes; Child quality of life

1. Introduction

Early development literature from the 1950s and 1960s generally treated rising per capita income as the central measure of economic development. Pioneering work by Kuznets [1] in the 1950s pointed to an inverted U-shaped relationship between per capita income and income inequality. By relating a country's per capita level of income to its income share of the top 20% of the population, Kuznets showed that if low-income developing countries strive to achieve higher rates of per capita gross domestic product (GDP), they would inevitably face

Socio-Economic Planning Sciences 34 (2000) 51±67

www.elsevier.com/locate/orms

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growing income inequality. Subsequent studies have attempted to test the validity of the Kuznets curve. Ahluwalia [2], Anand and Kanbur [3], and others noted that the expected inverse relation between rising GDP per capita and an income inequality measure seemed to disappear under more stringent multivariate statistical tests. Anand and Ravallion [4] showed that, when the extent of poverty and the level of public expenditures on basic services were included as explanatory variables, the presumed causal relationship between social indicators and per capita income disappeared.

However, the pendulum is swinging back with more recent studies by Ram [5] and Jha [6] showing a trade-o between growth and equity. Jha uses more recent estimates of income distribution obtained from the 1994 World Bank Social Indicators of Development database and attempts to validate the Kuznets curve. He ®nds that the trade-o between growth and equity holds, particularly when income growth rate and schooling variables are included in the regression. Ram [5] experiments with dierent functional forms and argues that a regression equation in which the constant term is constrained to zero increases the precision of the estimates and provides stronger support for the Kuznets hypothesis. The debate about the validity of the Kuznets curve is still ongoing as evidenced by the interchange between Ravallion [7] and the rejoinder by Ram [8].

In the past two decades, development specialists have increasingly recognized that pure economic indicators Ð such as GDP per capita or even income distribution measures such as the Gini coecients Ð do not suciently indicate overall welfare in a country. If the role of pure economic variables as indicators of welfare has become less clear, similarly, the debate continues about which social indicator best represents destitution and well-being [9]. Beginning with the World Bank's ``Basic Needs Approach'' in the early 1970s, to the United Nations Development Program (UNDP) Human Development Reports (HDRs), a range of social indicators has been combined with pure economic indicators in an eort to capture a country's quality of life in a single index.

The Human Development Index (HDI), is, as presented annually in the UNDP Human Development Report (see [10] for instance), one such index and by far the most widely used. Nevertheless, the HDI remains controversial. Some have gone so far as to label the HDI as a statistical artifact with limited policy value. Implicit in the HDI is the notion that it measures a country's quality of life better than a pure economic indicator such as per capita GDP. Ravallion [11] argues that the HDRs, by their undue concentration on the HDI, spend a considerable emphasis on distinguishing between ``good'' and ``bad'' growth and thus do not give sucient consideration to the overall growth and policy choices that favor high growth.

Luchters and Menkho [12] are particularly critical of the transformation of GDP values into human development values. They argue that the HDI income component is not sensitive to how income is denominated; that poor intertemporal comparisons are made when income is rising; and that income measures do not seem to follow the concept of diminishing marginal returns. From our perspective, the HDI can be criticized because the changes in output values occur without recognizing the changes in the resource base from which these outputs are derived. In this regard, Data Envelopment Analysis (DEA) is able to measure the eciency of countries in delivering a higher quality of life by explicitly emphasizing that output levels are accounted for by varying resource commitments.

More recently, Dasgupta and Weale [13] employed the Borda Rule (that is, ranks based on

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the sum of individual factor ranks) to rank order general well-being among the world's poorest countries. Following the approach of Dasgupta and Weal [13], Kotamraju [14] constructed a separate child quality of life (CQL) ranking for 22 less developed countries (LDCs). Kotamraju also developed a similar Borda Rule ranking for the 38 LDCs used in this study.

Unlike the traditional ranking approach to establishing a CQL index, the current research uses DEA to measure and rank the relative technical eciency by which 38 LDCs provide for childrens' future potential for, and quality of, life. Signi®cant social, cultural, and economic factors are treated as resources to be eciently rationed in order to maximize survivability, youth physical development, and literacy. The DEA approach ranks development eciency by evaluating the extent to which each LDC minimizes input components or conditions (the resources to be eciently rationed) and maximizes outputs or goals (survivability, youth physical development, and literacy). This approach is unique in that each LDC is allowed to select the particular set of importance weights or coecients (for the inputs and outputs) that allows the LDC to achieve its maximum CQL ranking. In short, a unique best production or transformation relationship can (and usually does) exist for each individual LDC. While conventional ranking schemes employ either equal, ®xed, or regression weights (thus implying a single function for all cases), DEA allows each particular LDC to choose the set of weights that maximizes its eciency in the face of the remaining LDCs, where the latter are constrained to employ that ``best'' set of weights.

There are several ways to measure the welfare of children. Often this is done with a single measure with the most common being the rate of child mortality. United Nations International Education Fund (UNICEF) [15] thus uses the under-®ve mortality rate as a measure for child welfare and ranks countries accordingly. In addition to the UNICEF measure, Kotamraju [14] added youth literacy rates and the percentage of children aected by chronic malnutrition. This CQL ranking procedure is comprised of these three equally weighted components. Other factors Ð as we present in this study Ð are also related to CQL. These include per capita real domestic product, population per doctor, female literacy rate, and female average age at ®rst marriage.

The current paper rede®nes the relationship between the above variables into an input± output or transformation paradigm. Three outputs or goals describe actual and potential child quality of life and are represented by (1) the under-®ve survival rate, (2) the lack of severe malnutrition, and (3) youth literacy rate. Inputs, resources, or conditions that determine actual or potential CQL can be represented by (1) per capita real domestic product, (2) female literacy rate, (3) females average age at ®rst marriage, and (4) population per doctor. This speci®cation of a transformation function leads to conclusions very dierent from those generated by contemporary ranking methodologies. We assert that many of the socio-economic indicators comprising well-known development indexes can be pro®tably rede®ned into resources to be minimized and goals to be maximized. In our proposed model, these inputs are thought to in¯uence the future quality of life Ð which is represented by outputs Ð of a developing country. This unique view of development indicators leads to a very dierent and more policy-oriented view of national development achievements.

It is interesting to note that an input±output transformation function is not even a necessary assumption to construct an index of rankings. A DEA-based CQL index could be constructed by maximizing indicators (variables) that would lead to a higher index ranking, and minimizing

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those indicators (variables) that are inversely related to the index ranking. The advantage of this procedure vs. the Borda Rule, or other ®xed weight schemes, is that the linear programming weights or coecients are explicitly chosen to maximize the individual country's ranking. With the DEA approach, weights are not equal nor subjectively selected, but are instead chosen by an optimization criterion. However, by formulating a reasonable transformation function, as we have with the CQL index, policy and program initiatives that maximize outputs, i.e., move the lowest ranking countries to higher ranks in the most ecient way, can be made more easily. A detailed account of the additive DEA model and the sensitivity analysis employed to rank the LDCs is presented in Appendix A.1 The following section summarizes the model measuring the eciency of child quality of life and describes the data set employed. We then present the ranking of the LDCs' eciency in improving CQL and discuss the usefulness of this approach.

2. The model and data

To evaluate the eciency in providing improved levels of childrens' well-being, a determination must be made of those cultural conditions and resource components (inputs) to be minimized and those CQL goal components (outputs) to be maximized. The ®rst goal indicator to be maximized is the under-®ve survival rate. International organizations usually express the under-®ve mortality rate (U5MR) as the percentage of children who die before reaching the age of ®ve. Hence, a DEA variable to maximized, SURVIVE, is de®ned as 100% minus U5MR; that is, the percentage of children who live beyond the age of ®ve.2

The second goal indicator to be maximized is the extent to which children thrive physically through the avoidance of malnutrition. The extent of malnutrition (STUNTING) is measured as the percentage of children under ®ve who are at least two standard deviations below the height-age norm. The DEA variable to be maximized, THRIVE, is thus de®ned as 100% minus STUNTING. (After surveying the athropometric literature, Dasgupta [9] concluded that any damage done to children occurs in the very early years (before the age of three) and ``growth failure in early childhood. . ._{predicts functional impairment in adults''.)}

The third goal indicator to be maximized is the youth literacy rate (YLR); that is, the percentage of literate children in the population age group 15±19. These goals Ð SURVIVE, THRIVE, and YLR Ð dier from the traditional income-centered approach that seeks to either maximize per capita income or minimize income inequality. In contrast, Anand and Ravallion [4] emphasize the human development approach to these goals, which ``. . ._{focuses on}

the (capabilities) of people Ð the lives they lead Ð not the detached objects they possess''. With respect to the input side of the transformation relationship, a whole host of economic

1

The additive model of DEA is uniquely appropriate to formulate indexes since it is neither input-oriented nor output-oriented only. Rather, the additive model simultaneously minimizes resources (inputs) while maximizing goals (outputs).

2

In the context of DEA, it makes little dierence if SURVIVE is speci®ed as an output to be maximized or, alter-natively, if U5MR is speci®ed as an input to be minimized. Even in a production model where pollution euents represent ``negative'' outputs, they can be treated as competitive inputs, and therefore minimized.

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and social factors can act as resources that in¯uence the previously identi®ed goals. First, income is recognized as a key variable. It expands capabilities directly through a greater command over material resources and leads to a lowering of poverty. Rising incomes and a more equal distribution of income should include improvements in the direct provision of public services. Higher average incomes are tied directly to the ®nancing of these public services. Anand and Ravallion [4] argue that higher incomes must be directed towards the poor in the form of increased public services. Thus, in the DEA framework, income is seen as an input into the development process that improves the potential of LDCs' children. In this study, per capita real gross domestic product (GDPCAP) is the income variable utilized and is de®ned in purchasing parity dollars.

For the second and third resource components, we look to the capabilities of women who are expected to raise children's welfare. According to the development literature, an improvement in the economic status of women increases income-earning capabilities of the individual and the family [16,17]. In addition, with the reduction of gender inequalities at home, the social status for women in society should also improve [18]. Hence, social and cultural conditions are expected to raise the welfare of children. The two variables chosen here as inputs for these considerations were female literacy rate (FEMLIT), which is the percentage of adult women who are literate, and female average age at ®rst marriage (FIRMAR).

Finally, avoiding morbidity and undernourishment are necessary for living well. And, so, for children, a critical factor in their survival is avoiding hunger and disease. Moreover, as Dasgupta [9] explained, better health ``. . ._{is in great part a matter of medical application.}

These considerations form the basis of the claim that in poor countries attention needs to be given to public health''. Therefore, to account for health inputs we use doctor per person (DOCPOP), which indicates doctor availability per capita.

With these inputs to be minimized and outputs to be maximized, the form of the proposed model is given as follows:

Inputs (minimize resource use or conditions): . per capita real domestic product (GDPCAP) . female literacy rate (FEMLIT)

. female average age at ®rst marriage (FIRMAR) . doctor per capita (DOCPOP)

Outputs (maximize outputs or goals): . under-®ve survival rate (SURVIVE)

. 100% minus STUNTING (THRIVE)

. youth literacy rate (YLR).

Extensive missing cases for these seven variables limited the population to 38 LDCs. The data on STUNTING (used to derive THRIVE), YLR, GDPCAP, DOCPOP, FEMLIT, and FIRMAR were obtained from the Human Development Report [10]. The State of the World's Children [15] contained data for U5MR (used to derive SURVIVE). Table 1 shows the data for the 38 LDCs on which the DEA was performed.

Based on these inputs to be minimized and outputs to be maximized, our model demonstrates that child welfare improvements should depend not only on material inputs, but

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Table 1

Child quality of life dataa

Country

Sudan 83.1 68 37 949 12 21.3 0.000098

Algeria 93.9 87 88 3011 46 21.0 0.000429

Jamaica 98.1 93 100 2979 99 25.2 0.000490

Mali 77.5 66 67 572 24 18.1 0.000043

Ghana 86.3 61 88 1016 51 19.3 0.000049

Bangladesh 86.7 35 46 872 22 16.7 0.000145

Cameroon 87.4 57 77 1646 43 17.5 0.000082

Thailand 96.7 72 99 3986 91 22.7 0.000159

Zambia 80.0 41 90 744 65 19.4 0.000140

Venezuela 95.7 93 97 6169 90 21.2 0.001429

Tunisia 94.2 77 95 3579 56 24.3 0.000463

Rwanda 81.1 66 65 657 37 21.2 0.000013

Mainland China 97.3 59 93 1990 62 22.4 0.000990

Burundi 81.9 40 80 625 40 20.8 0.000048

Jordan 95.4 79 97 2345 70 22.6 0.001163

Sri Lanka 97.9 61 96 2405 84 24.1 0.000181

Colombia 97.9 82 94 4237 86 20.4 0.000813

Morocco 90.9 66 80 2348 38 21.3 0.000210

Kuwait 98.3 86 82 15178 67 22.9 0.001450

Zimbabwe 91.2 69 81 1484 60 20.4 0.000139

Iran 93.8 45 79 3253 43 19.7 0.000339

Philippines 95.4 55 96 2303 90 22.4 0.000152

Dominican Republic 92.4 74 94 2404 82 20.5 0.000565

Egypt 91.5 68 65 1988 34 21.3 0.001299

Costa Rica 98.2 92 97 4542 93 22.7 0.001042

Mexico 96.3 78 96 5918 85 20.6 0.000806

Uruguay 97.6 84 99 5916 96 22.4 0.001961

Paraguay 94.1 83 96 2790 88 22.1 0.000685

Pakistan 86.6 40 50 1862 21 19.8 0.000340

India 87.4 35 66 1072 34 18.7 0.000397

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Table 1 (continued)

Bolivia 87.4 49 94 1572 71 22.1 0.000654

Panama 97.0 76 95 3317 88 21.2 0.001000

Chile 97.9 90 98 5099 93 23.6 0.000813

Peru 90.3 57 96 2622 79 22.7 0.000962

Ecuador 91.8 61 95 3074 84 22.1 0.001220

Nigeria 81.2 46 78 1215 78 18.7 0.000156

Guatemala 90.8 42 67 2576 47 20.5 0.000459

Brazil 93.3 85 92 4718 80 22.6 0.000926

a

Sources: UNICEF [15], SURVIVE from Appendix Table 1, UNDP [10], THRIVE from Table 11, YLR and FEMLIT from Table 5, GDPCAP from Table 1, FIRMAR from Table 8, and DOCPOP from Table 12.

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other social and health resources as well. Material inputs to a child's upbringing can be represented by the variable GDPCAP. Social and cultural factors like FEMLIT and FIRMAR are responsible for lowering child mortality and illiteracy. Similarly, the necessary health resources can be partially proxied by DOCPOP and also in¯uence survivability, physical thriving, and literacy. Within the DEA context then, an ecient country has comparable material, social, and health resources (to other countries) but better utilizes those resources to produce greater survivability, physical thriving, and literacy. Conversely, an ecient country may similarly be viewed as having comparable outputs of survivability, physical thriving, and literacy, but generally produces those levels of outputs with less material, social, and health resources.

3. Rankings of LDCs' eciency

Table 2 arrays the stability index values and yields stability index rankings from 1 (representing the most robustly ecient LDC) to 38 (representing the most robustly inecient LDC). It also gives the overall ranking based on the Borda Rule constructed in Kotamraju [14]. A casual viewing of the results does not indicate any obvious patterns amongst the outcomes. The Spearman rank correlation was ÿ0.024 but not statistically signi®cant at any level. Comparing CQL rankings using DEA weights and Borda rankings using equal weights, even with identical goals (outputs), resulted in an expected negative correlation. This is to be expected because LDCs with few resources and comparable outputs, when contrasted to LDCs with larger resource bases, are judged to be more ecient. But, from the Borda ranking, these LDCs appear to have somewhat less child development since no comparison to the resource base is made (see also Breu and Raab [19]).3 Hence, the Borda Rule appears to rank the absolute CQL, while the DEA index ranks the relative eciency in delivering CQL.

By organizing LDCs into ®ve continents and/or regions, a Kruskal±Wallis test statistic (H=17.61) rejected that the ®ve groups came from the same population at the 0.01 level (Chi-square critical value=13.28). Based upon these results, one could conclude that statistically signi®cant dierences exist in the robustness of eciency classi®cations across these geographic groupings. By computing pairwise dierences in average stability index ranks using Dunn's procedure of the Kruskal±Wallis rank test for dierences in pairs of medians [21], only two signi®cant comparisons existed at the ®ve percent level of signi®cance. Sub-Saharan African LDCs appeared more robustly ecient relative to South American and to Central American LDCs, which appeared as the most robustly inecient.

Table 3 displays the comparison of outputs and inputs for Sub-Saharan Africa versus South America and Central America. In general, South American output means are somewhat higher (SURVIVE by nearly 13%, THRIVE by 33%, and YLR by almost 30%), while resource means are considerably higher (GDPCAP by 306%, FIRMAR by 9%, FEMLIT by 87%, and

3

A U.S. News and World Report [20] ®xed-weight ranking of the quality of institutions of higher learning when correlated to a DEA ratio model ranking, resulted in a signi®cant negative correlation. The dierence exists because a DEA ranking is an eciency ranking, while the U.S. News ranking is a quality ranking.

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POPDOC [the reciprocal of DOCPOP] by ÿ95%). The negative sign on the POPDOC means implies that South American countries have more doctors per population unit. Similarly, comparison of the descriptive statistics for Sub-Saharan African versus Central American output means and input means yields similar ®ndings. Thus, Central American output means are somewhat higher (SURVIVE by over 14%, THRIVE by almost 33%, and YLR by 24%),

Table 2

Stability index rankings

DEA rank Borda rank Country Continent/Region Stability index value (y)

1 29 Sudan Sub-Sahara Africa 0.1469

2 15 Algeria North Africa 0.1209

3 1 Jamaica Central America 0.1001

4 30 Mali Sub-Sahara Africa 0.0898

5 27 Ghana Sub-Sahara Africa 0.0732

6 38 Bangladesh Asia 0.0669

7 28 Cameroon Sub-Sahara Africa 0.0660

8 6 Thailand Asia 0.0625

9 31 Zambia Sub-Sahara Africa 0.0521

10 5 Venezuela South America 0.0421

11 14 Tunisia North Africa 0.0407

12 32 Rwanda Sub-Sahara Africa 0.0367

13 19 Mainland China Asia 0.0349

14 34 Burundi Sub-Sahara Africa 0.0316

15 7 Jordan Middle East 0.0281

16 13 Sri Lanka Asia 0.0270

17 9 Colombia South America 0.0234

18 23 Morocco North Africa 0.0228

19 8 Kuwait Middle East 0.0222

20 22 Zimbabwe Sub-Sahara Africa 0.0191

21 26 Iran Middle East 0.0154

22 18 Philippines Asia 0.0133

23 17 Dominican Republic Central America 0.0125

24 24 Egypt North Africa 0.0122

25 2 Costa Rica Central America 0.0112

26 10 Mexico Central America 0.0106

27 4 Uruguay South America 0.0087

28 11 Paraguay South America 0.0052

29 37 Pakistan Asia 0.0047

30 36 India Asia 0.0041

31 24 Bolivia South America 0.0036

32 12 Panama Central America 0.0028

33 3 Chile South America ÿ0.0004

34 21 Peru South America ÿ0.0122

35 20 Ecuador South America ÿ0.0174 36 34 Nigeria Sub-Sahara Africa ÿ0.0282 37 33 Guatemala Central America ÿ0.0284

38 16 Brazil South America ÿ0.0334

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while resources are considerably higher (GDPCAP by 266%, FIRMAR by 5%, FEMLIT by 57%, and POPDOC [reciprocal of DOCPOP] by ÿ93%). Note that the standard deviation and the resulting coecients of variation in the material and physical inputs suggest diering resource endowments, even between countries of a particular continent. Notwithstanding some of these large dierences that make policy generalizations dicult, the general pattern of eciency rankings between Sub-Saharan Africa compared to South America and compared to Central America appear valid.

On a more intuitive level, we oer the following explanation. Given the much higher resource levels of South American and Central American LDCs, they are more ``inecient'' in delivering the desired outputs. In other words, despite their higher resource levels, South American and Central American countries are not doing a signi®cantly better job in taking care of their children than are Sub-Saharan African countries, who have comparable, if not somewhat smaller, levels of output (survivability, youth physical development, and youth literacy rates). Conversely, from the African point of view, a higher general level of deprivation requires subsistence countries to allocate, albeit at a very productive level, a signi®cantly higher proportion of their resources towards improving the lives of children in order to achieve a minimally accepted output. Hence, Sub-Saharan African countries are more ``ecient'' in fostering CQL.

A major problem faced in this paper is the lack of complete and comparable data at a point in time for an extensive set of countries. Only very recently has a concerted eort been made to collect such data for policy analysis. To arrive at a more policy-based CQL index, other policy inputs such as government expenditures Ð particularly those relating to the raising of health and literacy levels of poor children Ð should be included over the proxy variables

Table 3

Descriptive statistics for outputs and inputs (Sub-Sahara Africa vs South America and Central America)

Region/Continent SURVIVE THRIVE YLR GDPCAP FIRMAR FEMLIT POPDOC

Sub-Sahara Africa

Means (1) 82.3 57.1 73.7 990 19.7 45.6 20,339 Std. Dev. 5.9 11.7 16.0 388 1.4 20.5 21,515

C.V.a 5.0 20.5 21.6 39 7.1 40.0 106

South America

Means (2) 94.0 76.0 95.7 4022 22.1 85.2 1067 Std. Dev. 3.7 15.9 2.0 1592 0.9 7.7 341

C.V.a 3.9 20.9 2.1 39.6 4.1 9.4 32.0

Percent Dierence in Means (1)±(2) 12.8 33.1 29.7 306.3 9.0 86.8 ÿ94.7 Central America

Means (3) 95.5 75.8 91.5 3623 20.7 71.5 1358 Std. Dev. 3.1 18.5 12.8 1357 1.3 24.6 504 C.V.a 3.3 24.4 13.3 37.5 6.2 34.5 37.2 Percent Dierence in Means (1)±(3) 14.2 32.8 24.2 266.0 5.1 56.8 ÿ93.3

a

Note: C.V.=coecient of variation.

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included in our model. In addition, discretionary inputs such as female labor force participation rate may be built into the development of an improved CQL index. Finally, other policy-induced changes such as trade patterns, along with more autonomous ones such as weather, may be considered signi®cant conditioning factors of a more comprehensive CQL index [22].

Another problem exists when rankings are used as a production-policy paradigm, rather than a simple index composed of directly and inversely related indicators. A data problem appears because several inputs, especially domestic income and per capita doctors, are macro variables, which can account for goals and outcomes other than CQL. For example, doctors per capita may not be the critical input in improving health outcomes for LDCs (as many might believe). Mass immunizations and community health programs may thus be more ecient in the poorest of LDCs and data for such inputs need to be included. Proportion of median family income directly related to child care and per capita pediatricians or doctors devoted to child health care outcomes might be more acceptable inputs. Despite the limited set of variables used for developing the CQL index, our model begs for the inclusion of these additional variables to improve policies regarding child quality of life. However, the paucity of accurate data, even at the macro level, suggests that such re®nements are unlikely until the importance of such discretionary inputs is recognized.

4. Summary and conclusion

A CQL index for 38 LDCs, for which data exist, is developed and subsequently employed using a programming approach. A transformation relationship establishes the goals to be maximized Ð such as promoting child survival, avoiding failure to thrive, and raising child literacy Ð subject to resource-availability indicators (output per capita, doctors per capita) and conditioning variables (female literacy rate and female age at ®rst marriage) which are to be minimized. This approach better distinguishes between inputs and outputs and avoids using equal or subjective weights as employed in more conventional rankings of various economic and social development indexes. It allows a particular LDC to choose an optimum set of weights, i.e., one that maximize its eciency robustness. A sensitivity analysis not only allows the ranking of inecient LDCs, but also measures and ranks ``robustly'' ecient LDCs that comprise the eciency frontier. By focusing on eciency in the provision of CQL policy, and not the absolute level of CQL as might be included in, say, a Borda ranking, Sub-Saharan African LDCs are evaluated as more robustly ecient, while South American and Central American LDCs are found to be as more robustly inecient. This counter-intuitive result may occur because South American and Central American countries, with signi®cantly more resources, do not achieve comparably higher child development goals.

Subjective or ®xed weighted indexes measuring only outputs do yield some information suitable for policy planning. But, this is only ``half the picture''. The eciency approach used in this study considers the eective use of policy initiatives by emphasizing outcomes relative to the resource base employed and focuses on the eectiveness of these inputs. The obvious diculty in implementing this model, as presented, underscores the necessity of gathering information on critical policy variables, especially on the input side.

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Acknowledgements

The authors wish to thank Jean Jacobson for her editing assistance and Tabitha Schmidt for the computations.

Appendix A

A1. Data Envelopment Analysis

Data Envelopment Analysis (DEA) can be used to evaluate the relative technical eciency of transforming a set of inputs or resources to produce a set of multiple outputs. As a linear programming implementation of Farrell's [23] notion of technical eciency, DEA is an ``extremal'' approach to eciency evaluation. In particular, an ecient frontier is constructed that is composed of LDCs that either (1) use as little input as possible to produce a given level of output, or (2) produce as much output as possible from a given level of input consumption.4 Those LDCs meeting one of the above criteria comprise the ecient frontier and are technically ecient, while those LDCs not on the ecient frontier are technically inecient (enveloped by the ecient LDCs).

The original model of DEA, known as the ratio or CCR model [24], has been joined by other DEA models [25,26,33], including the additive model [27], the model of interest in this research. To rank order the LDCs from most robustly ecient to most robustly inecient using the additive model of DEA, we employ a two-step process. In the ®rst step, the technical eciency status (ecient or inecient) for each LDC is determined by solving a linear program for each LDC. This ®rst step serves only to categorize LDCs as either ecient or inecient. In the second step, dierent linear programs are solved for the ecient and inecient LDCs. With respect to a particular ecient LDC, the linear program yields a measure of its eciency ``robustness''; with respect to a particular inecient LDC, the linear program yields a measure of its ineciency ``robustness''. These robustness measures then comprise the index for rank ordering the LDCs. Below, we describe these two steps in greater detail.

In the additive DEA model, the observed input consumption and output production for a number of LDCs are measured. They are referred to as an LDC's component vector. All component vectors for the LDCs under scrutiny are combined to form the empirical production possibility set (PE):

vector of outputs and inputs, respectively, for LDCj.

4

Within the general DEA context, the entities or organizations under scrutiny are termed decision making units or DMUs. Our focus in this study is on evaluating the eciency of less developed countries or LDCs; therefore, in this appendix we substitute LDC for the more general DEA term, DMU.

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To determine the technical eciency status (ecient or inecient) for a given LDC, its component vector is compared to PE. If no component vector in PE, observed or hypothetical, can be found that strictly dominates the tested LDC, then the LDC is said to be technically ecient. Those LDCs for which a component vector can be found in PE that strictly dominates are said to be technically inecient. Fig. 1 provides a graphical depiction of a set of LDCs for a single-input, single-output example. From Fig. 1, LDCs Nos.1, 2, and 3 would be technically ecient, while LDCs Nos.4, 5, 6, and 7 would be technically inecient. Segments 12 and 23 comprise the ecient frontier.

Mathematically, the test for the technical eciency status of LDCj is achieved by solving the

following linear program:

min _ÿeT_Dÿ1

y s ÿe

T

Dÿ1x sÿ

s:t: Ylÿs _Y

j

XlsÿXj

eTlE1

l, s, sÿe0 2

where Y and X represent the matrices of the outputs and inputs, respectively; and s+ and sÿ denote the shortfall in production and excess consumption slacks, respectively. The eT vector is the sum vector, guaranteeing a convex combination or scalar multiple (less than one) of the LDCs under scrutiny.5 As the additive model is not units invariant, we note that Y and X are

Fig. 1. A hypothetical production possibility set and its frontier.

5_{The conventional additive model forces the}_l_{s to sum to unity (e}T_l_{=1), guaranteeing that the ecient frontier is} constructed of a convex combination of input and output levels. This constraint predetermines that an LDC with a unique minimum of any input must lie on the ecient frontier, regardless of how little output it may produce. By relaxing the lambda constraint as above (eTlE1), a particular LDC, even if it has a minimum of a given input, can be inecient. See Haag, Jaska, and Semple [28].

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transformed by component averages in the objective function (Dÿy-1, Dÿx-1) to assure common

results regardless of the units of measure chosen for each component.6

Again, the execution of (2) for each LDC serves only to categorize the LDCs as technically ecient or technically inecient. That is, the execution of (2) for each LDC does not yield a set of measures that can be used to construct a rank ordering. To develop a rank ordering of most robustly ecient to most robustly inecient LDCs, one additional linear program must be executed for each LDC, with the result yielding an 1-norm measure of the minimum distance to a Pareto optimum point (ecient frontier).

Charnes et al. [25,29] developed a sensitivity analysis technique based on the 1-norm measure of a vector. It de®nes the necessary simultaneous perturbations to the component vector of a given LDC that cause it to move to a state of ``virtual'' eciency. Virtual eciency is de®ned as a point on the ecient frontier where (1) any minuscule detrimental perturbation (increase in inputs and/or decrease in outputs) will cause an ecient LDC to become inecient, or (2) any minuscule favorable perturbation (decrease in inputs and/or increase in outputs) will cause an inecient LDC to become ecient.

For an ecient LDC, the 1-norm measure (herein termed stability index) de®nes the largest ``cell'' in which all simultaneous detrimental perturbations to the input and output components will not cause a change in the eciency status from technically ecient to technically inecient [32]. As such, the larger the stability index, the more robustly ecient the LDC is said to be. Those ecient LDCs with small stability indices will thus become technically inecient with smaller detrimental perturbations than those ecient LDCs with larger stability indices. Mathematically, the stability index for an ecient LDC (LDCj) is determined by solving the

following linear program:

min _y

s:t: Y E

lÿsyd0Yj

XE_l_sÿ_ÿ_y_d IXj

eT_l₁

l, s, sÿ, ye0 3

where y represents the stability index. The matrix of outputs and inputs are represented by Y(E) and X(E), respectively, with the component vector for ecient LDCj omitted. Finally, d0 and d1 are given by dT0 and dT1=(1,1,. . .,1), which cause y to simultaneously increase inputs

6

Haag, Jaska, and Semple [28] ®rst introduced the notion of pre-scaling the data by component averages to create a units invariant model. See Haag and Jaska [30] for a complete numerical analysis and explanation. See Lovell and Pastor [31] for an alternative pre-scaling technique that must be used when the data contain zero (0) and/or negative values.

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and decrease outputs as the linear program determines the optimal solution. Fig. 2 provides a graphical depiction of the1-norm measure (stability cell) for ecient LDC No. 2.

For an inecient LDC, the stability index de®nes the largest ``cell'' in which all simultaneous favorable perturbations to the input and output components will not cause a change in the eciency status from technically inecient to technically ecient. As such, the larger the stability index for an inecient LDC the more robustly inecient the LDC would be. An inecient LDC with a large stability index thus rests a greater distance from the ecient frontier than does an inecient LDC with a smaller stability index. Mathematically, the stability index for an inecient LDC (LDCj) is determined by solving the following linear program:

max _y

s:t: Ylÿs _ÿ_y_d

0Yj

XlsÿydIX_j

eT_l₁

l, s, sÿ, yr0 4

Fig. 2. Dashed line within stability cell represents theymeasure for LDC No. 2.

Fig. 3. Dashed line within stability cell represents theymeasure for LDC No. 7.

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where all notations are de®ned in the prior formulations. Observe that y simultaneously decreases inputs and increases outputs as the linear program determines the optimal solution. Fig. 3 provides a graphical depiction of the1-norm measure (stability cell) for inecient LDC No. 7.

Once the stability index is known for each LDC, the LDCs can be ranked from most robustly technically ecient to most robustly technically inecient. To do so, the stability indices for inecient LDCs are ®rst negated. Then, the LDCs can be rank ordered from highest to lowest based on their stability index values.

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