Equity in the Education System and Student Effort: A lesson from Korea
for Poverty Reduction in Colombia
Presented by:
Sergio González Carrillo
Thesis Advisor:
Shin Soon-Ok
Master’s in Economics
Economics Faculty
Universidad de los Andes
Abstract
Poverty reduction is a major challenge for Colombia. Considering the two main drivers of poverty reduction, economic growth and equity, the education system plays a key role and offers an opportunity for the design of an effective strategy for poverty reduction in Colombia. In countries such as the Republic of Korea, equity in the education system appears to have contributed to the attainment of high growth rates coupled with low economic inequality through fast and well-distributed human capital accumulation. Due to its importance on individual educational outcomes, student effort is a variable that can be decisive for a comprehensive explanation of the remarkable performance of the Korean education system. The mechanisms through which particular characteristics of the education system can affect the determination of aggregate levels of student effort has been, however, an understudied subject in the economic literature. Based on this fact, the aim of this thesis is to contribute to this undertaking by proposing a basic theoretical model, which allows the analysis of the impact of school input inequality on the overall efficiency of the education system. The explored mechanism is endogenous student effort. The main result is that higher disparities in school inputs can affect negatively the equilibrium level of student effort for all socioeconomic strata. Consequently, it is possible to conclude that equity in the education system would not only stimulate social mobility and a reduction in economic inequality; it would have a direct impact on the efficiency of the education system through its positive impact on student effort. This main result implies that equity in the education system can boost human capital accumulation and consequently economic growth through a previously unexplored mechanism. Due to its potential to address simultaneously economic growth and equity, its implementation where there are high disparities in school inputs is promising as a main component of an effective poverty reduction strategy.
Table of Contents
1. Introduction ... 4
2. Literature Review ... 6
2.1. Poverty reduction and challenges for Colombia ... 6
2.2. The education system: Opportunities for lessons in poverty reduction ... 11
2.3.The role of education in the Korean miracle ... 14
2.4. Economic research on student effort, education and social mobility ... 18
3. Construction of the Basic Analytical Model ... 20
3.1. Elements of the basic model ... 21
3.2. Description of the core problem ... 22
3.3. Maximization and results ... 25
3.4. Interpretation and implications of the main result ... 28
3.5. Validating exercise ... 33
4. Conclusions and Further Research ... 35
1. Introduction
Colombia faces great challenges on its struggle for poverty reduction. The complexity of its causes demand continuous efforts in policymaking, constant improvement and great deals of creativity. In the pursuit of strategies to effectively address poverty reduction, the academic community can make valuable contributions by providing useful information for the pertinent policymakers. This thesis is the result of such an effort, as it was developed with the aim of contributing to the discussion of one of the main problems faced by the Colombian society.
In this regard, the socioeconomic transformation that has taken place in East Asia during the last 6 decades is a natural source of valuable lessons for poverty reduction. Despite the fact that replication of a particular model in different historical, economic, cultural and political contexts is impossible in practice, focusing on the key aspects of successful experiences may prove to be useful for this undertaking. The particular case of the Republic of Korea and the impressive transformation it has undergone in the last 6 decades offer an especially valuable experience. In this short time frame, this country evolved from having an economic and social condition worse than in most South American nations (Kay, 2002, p. 1084) to become the twelfth largest economy in the world by purchasing power parity (International Monetary Fund, 2012) and a world player in the high technology sectors. Throughout this process, absolute poverty has been almost completely eradicated, as high economic growth has been coupled with relatively low levels of economic inequality1. The analysis of the economic history of the Republic of Korea suggests that the education system has played a dual role. On the one hand, it was able to sustain high economic growth by attaining exceptional rates of human capital accumulation, especially at the primary and secondary levels (Lee J.-W. , 2007, p. 7). However, despite the efforts of experienced researchers, it is still uncertain how did the Republic of Korea and other Asian countries achieve such impressive educational attainments2. On the other hand, it guaranteed a relatively equitable distribution of human capital, which was decisive to prevent
1 This is a disputed statement. Further discussion on this issue is presented in page 15.
2For example, Lee & Barro in 1997 acknowledge having been unable to account for the superior performance of
East Asian education systems. 10 years later, the former author stated that “It is quite puzzling how Korean students perform better than others” (Lee J.-W., 2007, p.7) and discards spending in education as a sufficient explanation. Subsequent analysis such as the one presented by Lee in 2010, support this last observation.
economic inequality from rising in a political context where distribution issues were not a primary concern3. By affecting and shaping the dynamics of both economic growth and economic inequality, the characteristics of the education system of Korea can be of great interest as a source of valuable lessons for the struggle against poverty in Colombia.
With this in mind, the main contribution of this thesis is to present a simple theoretical model able to describe the impact of certain features of the national education systems on their outcomes. The model has been inspired by the observation of special characteristics of the Korean education system such as high competition and effort levels, the historically enforced policy of egalitarianism, and its contrast with the Colombian education system. By considering students the decision-making unit, it will allow to analyze the impact of school input inequality on the efficiency of the education system through an analytical category that has been generally overlooked by the economic literature: student effort. This mechanism implies that school input inequality has negative effects on total human capital accumulation, not only on social mobility and economic inequality. If this link could be demonstrated, it would have considerable implications on the design of poverty reduction strategies for Colombia and other countries.
This thesis is structured as follows. Section 2 presents the literature review. First, it introduces the main challenges Colombia faces on its struggle for poverty reduction, focusing on the two main drivers of poverty reduction: economic growth and reduction in economic inequality. Afterwards, it presents a discussion on the special role of the education system and points out its value for poverty reduction through these two main drivers of poverty reduction. The following subsection introduces an overview of the impressive transformation of the Republic of Korea, and points out the importance of the education system in this process. In this subsection, student effort is identified as a relevant category that seems to be decisive for the educational achievements of the Republic of Korea. Finally, section 2 reviews the existing economic literature on student effort, and on the links of education and intergenerational social mobility, a key determinant of persistent economic inequality. Section 3 deals with the construction of the theoretical model. It presents the basic equations, describes the core problem, and finally presents the main results and their interpretation. Finally, it proposes a simple econometric exercise that could be carried out in order to validate the model and its main
3 As Lee (2011, p. 16) has pointed out, there is a general agreement on the veracity of this statement. It is disputed, however, whether the Korean education system has the same contribution to economic equity today.
predictions if the required information was available. In section 4, conclusions are presented along with a discussion of the limitations and pending issues to be addressed in further research.
2. Literature Review
2.1. Poverty Reduction and Challenges for Colombia
As poverty reduction is the main concern of this thesis, it must be recognized that the concept of poverty is not unique, as several definitions have been proposed. Among these definitions, the most traditional has been poverty in absolute terms, directly related to human material well being as portrayed in the following definition: “The state of one who lacks a usual or socially acceptable amount of money or material possessions. Poverty is said to exist when people lack the means to satisfy their basic needs…” (Encyclopedia Britannica, 2012). There are also alternative concepts of poverty that were developed to address the perceived shortcomings of the focus on incomes or resources. Probably the most influential is the one proposed by the capabilities approach, which recognizes the social, cultural and individual particularities that constraint the ways in which material limitations affect human well-being. Nonetheless, for the sake of simplicity and avoiding deviation from the discussion, we will focus on the traditional concept of poverty: absolute deprivation of material resources necessary to satisfy basic human needs.
By understanding poverty in this straightforward way, we can easily conclude that the two most important conditions required for a successful and continued reduction in poverty are: sustainable and persistent economic growth, which provides the means for increasing individual incomes; and the transition to a more equitable distribution of incomes, which is capable of raising the relative incomes of the poor. These two conditions have been widely recognized as the two main drivers of poverty reduction (Atlinc, et al., 2006, p.2; Kwasi, 2010, p.17-18). Despite this, economic history has proven that these two conditions have been very difficult to attain for developing countries, let alone simultaneously.
Regarding these two conditions, in recent years, Colombia and Latin America as a whole have had an acceptable performance in terms of the trends in both economic growth and distribution. Economic growth after the 1998-2002 period –known as the lost half decade– has
been higher than world average, reaching an average growth level of nearly 5%, a situation that has helped the region regain the economic weight it lost during the 80s (OECD/Economic Commission for Latin America and the Caribbean, 2012, p.13). Such outstanding growth levels were made possible by diverse factors, both internal and external (Economic Comission for Latin America and the Caribbean, 2012, pp. 24-26, 29-31). Evolution of economic distribution has also been positive in this period. However, there is evidence that these improvements have been unequal throughout the region. In addition, most of them appear to be the result of political tendencies: a shift towards left wing governments in many countries in the region, and growing social expenditure from right wing governments (Birdsall, Lustig, & McLeod, 2011, p. 8). Social expenditure has been increasingly targeted towards internationally recognized policies, among which the most significant examples are probably the conditional transfer programs, which have been implemented in many countries in the region such as Brazil, Mexico, Peru, Chile, Panama, Honduras, Guatemala, Nicaragua and Colombia (Fizbein & Norbert, 2009). Despite this progress, as of 2012 the region maintains its position as the most unequal on earth (Economic Comission for Latin America and the Caribbean, 2012), followed closely by Subsaharian Africa.
As a result of the developments in both economic growth and inequality, poverty in the region has been reduced from 44% in 2002 to 33% in 2009, and has continued to fall (Ferreira et al., 2012, p. 1). However, the same authors note that 66% of this reduction is explained by pure economic growth, with the remainder explained by falling income inequality. In addition, the bulk of the remaining 34% is the result of falling inequality in a very limited subset of countries: Brazil, Argentina, and to a lesser extent, Chile and Peru. Conversely, poverty reduction in Colombia has been almost completely driven by economic growth (Ferreira et al., 2012, p. 4). According to official data using ‘old’ methodology (Departamento Administrativo Nacional de Estadística, 2012; Nova, 2012) –an absolute poverty measure– poverty has been reduced in the last decade from 53.7% in 2002 to 50.3% in 2005. In subsequent years, it was further reduced to 45.5% in 2009, 39.8% in 2011 and 37.8% in 2012. By comparison, this progress is somewhat lower to the one achieved in the broader region4. Considering that the growth-elasticity of poverty reduction is lower for countries with higher inequality (Perry & López, 2008, p. 2), that is, more growth is required to achieve an equivalent reduction of poverty; this result is disturbing
4 While from 2002 to 2009 poverty in Latin America was reduced from 44% to 33% (11 percentage points). In Colombia, poverty was reduced from 53.7% to 45.5% (8.2 percentage points).
but expected.
Regarding the trends of the first driver of poverty reduction in Colombia, economic growth, we can appreciate that these are similar to those of Latin America as a whole. The general outlook is positive. Economic growth has remained in relatively high levels despite the crisis in the developed world, as in the period 2002-2007 it grew at an average rate of 5.9%. In 2008 and 2009 at 2.5%, which then recovered to 4% in 2010, 6.3% in 2011 and 4% in 2012 (DANE, 2013).
On the other hand, one of the most outstanding challenges for poverty reduction in Colombia is the extremely high and persistent economic inequality. As of 2012, Colombia was the third most unequal country in the region, measured by its Gini Coefficient (0.58 and 0.60 for its capital city), with only Guatemala and Honduras exceeding this value (UN-Habitat, 2012, p. 45). This situation has engendered, among other consequences, higher poverty and extreme poverty levels than other countries with comparable GDP per capita and development levels. In addition, evidence suggests that economic inequality has not ceded much in recent years, as it has occurred in Latin America as a whole (Meléndez & Ferreira, 2012, pp. 34-35). A high level of persistent unemployment has made the reduction of inequality more difficult (Rhenals M., 2005, p. 49). These characteristics of the Colombian economy and society place inequality and the problem of inclusive growth as one of the key issues to focus in order to address poverty reduction.
Explaining the extreme inequality of both the region in general and Colombia in particular and the disaggregation of its components is a very difficult task. Economic researchers have identified a myriad of causes and their complex interactions5. In addition to these, Bourdieu & Passeron (1990) have argued from a sociological point of view that in addition to its role of transmission of information and human capital formation (and distribution) as viewed by most economists, the education system has the role of reproducing the social structure by inheriting privileges and other complex social mechanisms. This theory is of special interest for this thesis as it links particular forms of inequality in the education system with the levels of social mobility. Another very difficult task is to measure the complex relation between inequality and growth. As these two are the main drivers of poverty reduction, their relation is very important and should be taken into account when considering poverty reduction measures or strategies.
5 For example, Engerman & Solokoff (1994) and Acemoglu, Johnson, & Robinson (2001) among many others have contributed to this understanding by providing economic political and historical explanations.
Most economists have traditionally stressed the ‘equity and efficiency’ tradeoff in many contexts, advocating that policymakers should prevent social policies from hampering efficiency and economic growth. In addition, researchers from very diverse ideological origins ranging from Post Keynesians (Pasinetti, 1962) to Neoclassics (Carroll & Kimball, 1996) have described a channel that links inequality to higher economic growth. The general mechanism described is that richer individuals tend to have higher savings (and investment) ratios, and consequently higher inequality leads to higher aggregate savings and investment ratios. However, when examining the aggregate effect of inequality on savings and investment, considering indirect effects such as financial constraints, redistributive policies, and other phenomena, the net effect of inequality on aggregate savings and investment become theoretically ambiguous. Recent empirical research suggests that there is no evidence of inequality producing systematic effects on aggregate savings and investment ratios (Schmidt-Hebbel & Servén, 2000, p. 442).
More recently however, researchers have pointed out the negative effects of inequality on growth and economic efficiency, and its influence has increased significantly as demonstrated for example, by the Economic Commission for Latin America and the Caribbean –ECLAC (2012) report, the World Bank 2006 Report (Atlinc, et al., 2006), and the World Bank 2012 Conference on Equity. The channels through which inequality has been acknowledged to affect growth negatively are varied and very complex. Perry and López (2008, pp. 2-3) offer interesting explanations and discussions of the following channels: The ‘political economy’ channel, which operates as economic inequality generates political pressure for redistributive measures. In turn, these require taxation of economic activities affecting return rates, international competitiveness, and as a consequence capital accumulation. The ‘sociopolitical instability’ channel operates as societies with higher levels of economic inequality tend to have higher levels of intolerance, crime and violence. As a result, economic activities are negatively affected through the interrelated phenomena of increased expenditure in unproductiveactivities (such as security and defense) and rent seeking behavior; destruction of physical, human and social capital; perceived insecurity of property rights; and higher levels of distrust, which generates higher transaction costs. The ‘credit constraint’ channel operates as aggregate human and physical capital accumulation is affected by the fact that the relatively deprived are subject to credit constraints. This situation generates inefficiencies through the inability of such individuals to make efficient medium and long term economic decisions which manifests through low investment in education,
ubiquity of low capital informal economic activities, and in general, loss of economic opportunities. As a result, an effective strategy for poverty reduction in a context of high economic inequality such as in Colombia, should take into account the mechanisms described above. Measures able to stimulate economic growth and reduce economic inequality simultaneously should be prioritized as part of an integral poverty reduction strategy.
Recent political developments have revealed that political will to strengthen poverty reduction efforts in Colombia has increased. In recent years, important institutional reform that has placed poverty reduction and alleviation efforts at the core of public policy, have been undertaken. The institution that coordinates public efforts for social inclusion and reconciliation – social policy– is the Administrative Department for Social Prosperity (Departamento Administrativo para la Prosperidad Social in Spanish), which was created in 2011 replacing the
Presidential Agency for Social Action and International Cooperation (Agencia Presidencial para
la Acción Social y la Cooperación Internacional in Spanish) created in 2005. Innovative institutional arrangements such as the National Agency for Overcoming Extreme Poverty
(Agencia Nacional para la Superación de la Pobreza Extrema - ANSPE in Spanish), specifically designed to execute measures of extreme poverty reduction and alleviation; and the Special Administrative Unit for Attention and Integral Reparation of Victims (Unidad Administrativa Especial para la Atención y Reparación Integral de Víctimas in Spanish) specially designed to address the needs and issues concerning the victims of the Colombian Conflict –one of the most vulnerable sectors of the society–, were created in 2011. These institutions have contributed to strengthening institutional capacity for the implementation of social policy in general and struggle against poverty in particular. The chief programs of this nature include conditional transfer programs such as Families in Action (Familias en Acción in Spanish), a broad land restitution program for the internally displaced population and for the victims of the Colombian Conflict, cross-subsidies in public services, mentoring networks (for example, the UNIDOS network), focalization and preferential access to the state services, among others. In any case, only time can tell if this new institutional architecture would be able to address effectively the current and future challenges and boost poverty reduction. However, it reaffirms that political will is increasing, which is vital for success.
As it has been stated before, focus will be set in East Asia, as this region has outperformed Latin America during the last 6 decades, starting from a similar economic base. Just like Latin America, the economic, social and political systems are subject to great heterogeneity. Nonetheless, it is possible to trace some of the common aspects implemented by the most successful cases of East and Southeast Asia, such as the Republic of Korea, Taiwan, Hong Kong, Singapore, Malaysia, and more recently Thailand and China. In addition to the general preservation of macroeconomic stability and structurally high investment and savings ratios, two of the most recognized aspects are: i) the role of industrial policy in the attainment of strategic structural change and the development of export oriented economies and ii) the role of fast and effective human capital accumulation through the education system. An active and wide ranging industrial policy is one of the elements that is common to latecomer East Asian economies specially for some of its most outstanding cases such as South Korea, Taiwan, Singapore, China, and even to the earlier case of Japan. Although consensus about the positive or negative impact that industrial policy had on these cases, it is definitely one of the constituting elements of the East Asian model (Yoon, 2001), with the notable exception of Hong Kong.
On the other hand, the role of the education system on the accumulation and distribution of human capital and ultimately on economic outcomes constitutes a major and outstanding element in the East Asian success stories. The alignment of education and industrial policies, strong links between the academic and private sectors, along with efforts from the Government, families and students has generated the education boom that pushed human capital accumulation to unparalleled levels. Even though this important element of the East Asian success story has been recognized and emphasized in a practically unanimous manner, researchers have focused most of their efforts mainly on the role of the implementation of export-oriented strategies, high investment and savings ratios among others (Lee & Francisco, 2010, p. 1). The existing literature on the educational achievements of Asian countries during the past 6 decades point out the following factors as the main determinants: i) high rates of economic growth, ii) an emphasis on the policies needed to promote high economic growth, iii) the rapid transition from high to low fertility rates, and iv) strong public institutions (Lee & Francisco, 2010, p. 1). In addition, these same authors emphasized the importance of culture in educational outcomes, as there is evidence
that in the case of East Asia, it has had a statistically significant impact (2010, p. 14)6. Moreover, income inequality has also been demonstrated to affect educational outcomes, through credit constraints and other mechanisms (Flug, Spilimbergo, & Wachtenheim, 1998, p. 17). As a result, low levels of income inequality in East Asia provided a favorable environment for the achievement of successful educational outcomes. Among Asian countries, the Republic of Korea, Taiwan and Malaysia were the countries that had expanded most significantly their educational attainment between 1970 and 2010. Measured by total average years of schooling, increases in educational attainment for the population aged 15 and over have reached 5.36, 5.27 and 6 years respectively. Furthermore, the Republic of Korea has been the leader in total educational attainment with 11.74 total average years of schooling (Lee & Francisco, 2010).
On contrast, an overview of education in Latin American countries, such as the one offered by the recent report by the Inter American Development Bank (Bassi et al., 2012), confirms that there are enormous challenges for the improvement of national education systems. Efficiency and efficacy of investment in education are deeply affected by, among other causes, extremely high dropout rates. According to the same report, the average level in Latin America for secondary education is 46%. Inequality in educational attainment among social classes has reached severe levels in the region, as the average difference between the first and fifth quintiles has surpassed 7 schooling years, without even considering quality differences. In addition, according to the Program for International Student Assessment (PISA)7 test results in 2009 (Bassi et al., p. 69), the differences in measured abilities between the best and the lagging a student is much higher than in OECD and Asian countries – for all education levels. The exacerbation of these differentials is generated by the severe input quality differences that exist among schools for higher and lower social classes in the region. In Colombia, these differences are extreme. Even total class time per day is different for income levels, due partly to the existence of the “doble jornada”, a system created to expand gross coverage in a context of budgetary restrictions in which subsidized public schools receive two set of students per day, offering fewer class hours
6 For instance, (Lee & Barro, 1997, pp. 25, 30) show that an East Asia dummy variable is statistically significant in their cross-county comparison of educational outcomes. This can be a reflection of the importance of culture as Lee & Francisco recognized (2010). However, it also allows for the possibility that it could be the reflection of a missing variable such as student effort.
7 PISA test results are one of the most influential information sources for cross-country comparisons of educational attainment. These are carried out every three years starting from year 2000. The subjects of study are 15-year-old students from public and private schools. The 3 analyzed dimensions are science, mathematics, and reading skills.
per day than private schools for middle and high-class students. Recent evidence suggests that there are severe differences in educational outcomes from these schools (Bonilla, 2011), which further aggravates quality differences resulting from financial resources, teacher quality and other key inputs. Recent research has shown that Colombian gross coverage at all levels has expanded significantly in the last few years, which reflects an important institutional and financial effort. However, there is evidence that indicates that this growth has been achieved at the expense of
quality, especially in low class public schools (Barrera-Osorio, Maldonado, & Rodríguez, 2012). This shows that even though there are important programs to strengthen quality in official schools8, efforts have been insufficient vis-à-vis the enormous requirements and challenges. The policies required to improve quality, such as the elimination of the doble jornada, improved teacher policy, continuity and expansion of successful policies, among others, important financial efforts must be assumed, which implies that political will is crucial to advance towards a substantial improvement of the education system in Colombia.
Finally, the PISA test results suggest that the average performance of Latin American students is significantly below world average, and the same occurs to the top students in the region (in the case of Colombia, a mere 0.1% of the students surpassed the average performance of the world leader in the mathematics PISA results, Shanghai-China) (Bassi et al., 2012, p. 68). This last observation implies that not even the most privileged sectors of the Latin American society are being well prepared to compete effectively in a globalized world, despite the substantial resources invested on their education. That is, there are severe limitations and problems in the education system that cannot be traced merely to resource limitations, which affect the capacity of the educational system to accumulate valuable knowledge and abilities in students9.
As a result of what has been discussed earlier, we are able to conclude that: i) education has been extensively recognized to affect social and economic outcomes through various channels, ii) it constitutes a mechanism both for the transformation or the reproduction of the social structure and consequently iii) it constitutes one of Colombia’s main opportunities (and challenges) for
8 Some of the most recent programs that have been implemented in recent years include the following: Conditional transfers programs (among others, Familias en Acción, 2002-) vouchers for private education at the secondary level (PACES, 1991-1997), fee reduction programs (Gratuidad, 2008-), concession schools (1999-), among others. 9 This constitutes additional evidence that a key variable has not been taken into account. It is highly unlikely that total human and financial resources invested on the education of the average Shangainese is sufficiently higher than those allocated to the Latin American elites, causing such alarming differences in the 2009 PISA test results.
poverty reduction.
2.3. The role of education in the ‘Korean Miracle’
The education system of the Republic of Korea is the source of important lessons for the education systems of the rest of the world, as it has supported the transformation of the Korean society as the most educated in East Asia. First of all, high social (public and private) investment in education has been a major feature of the Korean society even during the Korean War, enduring throughout the next decades in which the ‘economic miracle’ took place, and currently continues to be a major obsession of both Government and families (Lee J., 2011, p. 1). Second, this country managed to become one of the world’s most developed and prosperous nations as well as a major international player with a knowledge-based economy, having started from an extremely low base. For example, as close as in 1960, according to the World DataBank, nominal GDP per capita in current prices was as low as US$155, about 62% of the Colombian equivalent of US$252 (The World Bank, 2013). And last but not least, Korea managed to carry out its economic development process while simultaneously maintaining a very low level of economic inequality, even mildly lowering it during the period, with the notable exception of the East Asian Financial Crisis (Fields & Yoo, 2000)10. This behavior of the Korean economy, defiant of the deeply entrenched Kuznets hypothesis, has an exceptional value for Colombia and particularly for its efforts to achieve high economic growth while lowering economic inequality.
Explaining the ‘Korean Miracle’ is not easy at all, considering scholars from Korea and abroad are far from reaching an agreement. Since this is a highly disputed issue, debates about its determinants have not been settled, including whether its example can be replicated by developing countries. A thorough and complete review of the debates and discussions around the Korean success greatly exceeds the scope of this article, and is of little use to our purpose. One of the few broad agreements is that education policy played a significant role both as an instrument of economic growth and development, as well as in stimulating social mobility, along its development process (Lee J.-W. , 2007, pp. 10-14) and (Lee K. , 2010, p. 58).
10 This is a controversial point of view. There are opinions according to which the “growth obsessed” pattern followed by the Republic of Korea deliberately neglected the dimensions of stability and equity. These are based on different sets of income distribution data, which show a moderate increase in the Gini Coefficient from 0.34 in 1965 to 0.4 in 1979 (Jeon, 1995, p. 73). In any case, during and after the high growth period of the Korean economy, inequality remained very low for international standards, despite “growth obsessed” economic policies.
A simple overview of the role of education on the different stages of the Korean development process in which there is a general agreement (although there are some differences in the classification of stages and its particular characterizations), as offered by Lee J.W. (2007, pp. 2-5) and Lee K. (2010, pp. 58-60) can be summarized as follows:
In the period from 1945 to 1960, that is, since liberation from Japanese rule to the arrival of President Park’s authoritarian rule, there was a major increase in primary education and a substantial increase in secondary education, resulting from aggressive Government programs and efforts such as illiteracy eradication. Korea managed to attain such accomplishments despite suffering the devastation generated by the Korean War (1950 to 1953). Both WWII and the Korean War left the economy almost completely deprived of physical capital. As a result, during the reconstruction period (1953-1962) and after the rise to power of President Park in 1962, the most abundant factor of production was unskilled labor, since Korea is not endowed with plentiful natural resources. This particular situation provided the economic conditions for the strategy to invest on primary education to maximize static and dynamic efficiencies simultaneously. This fact, coupled with an export-led growth strategy and high saving and investment ratios (with support from Official Development Aid in the first years) allowed Korea to accumulate physical capital at a high rate. This took place as a Hecksher-Ohlin trade pattern developed during the first stages of its development, as it focused on exporting light manufactures, intensive in unskilled labor. At around 1973, as the Government shifted economic strategy to focus on the heavy and chemical industries (HCI), the education strategy then shifted to emphasizing expansion of secondary and vocational education. In turn, this generated qualitative changes to factor endowments, supporting the structural change process towards these sectors, led by industrial policy. Indebtedness, excess capacity and international turmoil engendered economic crisis at the end of the decade of the 70’s, however, structural change was not reversed by the crisis. Around 1980, the assassination of President Park and the shift towards a more indirect leading of the economy, education was focused to deliver higher quality education and higher education. This provided highly qualified workers and technicians to the increasingly knowledge-intensive sectors, in the pursuit of strategic “constant upgrading” (Yoon, 2001, pp. 225-227).
On the other hand, evidence shows that the role of the education system on the preservation of low inequality levels through the successive stages of development –and in the presence of a
broad economic policy that favored growth over distribution– was decisive. The first element of the Korean education system that is recognized to be important in this sense is the principle of
egalitarianism, which was a fundamental element of the Central Education Policy until the 1990
reform (Lee S.-J., 2008, p. 99). Even though the reasons for building a largely egalitarian education system were not based on distributive concerns, but more so as a means for ensuring transparency and reducing competition among families for entry in the top high schools (Lee S.-J., 2008, p. 99); the result was a system that supported high social mobility, and guaranteed equality of opportunity to a great extent. This principle manifested through various features and mechanisms of the schooling system. One of these was random matching between students and schools, which guaranteed equal education opportunities for all social classes, and ensured interaction among students from different socioeconomic backgrounds: “The MOE [Ministry of Education] created a lottery system in 1968 by which students were randomly assigned schools in large school districts that were designed to include both wealthier downtown areas and poor outskirts of cities” (Seth, 2005, p. 6). Another very important measure that conveys this principle is the High School Equalization Policy, which was implemented in 1974. This policy aimed to equalize curricula, fees and teacher salaries for all high schools in the country, in order to standardize input quality, and to avoid private schools for students from wealthy families to hoard the best teachers available. All these measures have in common that they contribute to weaken the link between family background and social class to educational attainment and consequently, to individual economic outcomes. As a result, the educational system becomes an effective mechanism for social transformation, rather than social reproduction.
Now that it is clear how the Korean education system has stimulated social mobility, an equally important question arises: How did Korea manage to reach such high levels of human capital accumulation? The first explanation that comes into mind is political decision, which roughly translates as total government expenditure on education as a percentage of GDP. This explanation is partly correct, since we have learned that Korean policymakers designed and implemented education policy as a key element of an overall industrial policy. However, statistical evidence shows us that its levels were not high enough to explain the impressive levels of human capital accumulation, both measured as broad coverage (enrollment rates), but
especially in terms of the value of human capital accumulation (Lee K., 2010, pp.50-52). The second answer is high private expenditure in education, which is a consequence of the conditions
that engendered a structurally high demand for education services. Since on one hand, Japanese rule, the Korean War and the early 50’s land reform had destroyed the existing social class structure, and on the other, the Korean society was traditionally very homogeneous in culture, language and race, education became virtually the sole means of social advancement, and consequently became an obsession for all the sectors of society (Lee J.-W., 2007, pp. 8-9). In addition to this statement, this author also provides statistical evidence that the ‘quality of investment in education’ was kept high during the years in which expansion took place. However, the definition of ‘quality of investment in education’ is not very clear, and it seems to be sufficiently wide as to allow for various elements ranging from teacher and other input quality, alignment of investment projects with the general economic strategy, and good targeting of expenditure, as we learned from Lee J.W. (2007) and Lee K. (2010). The former author provides evidence of what could be considered teacher quality by showing us that as early as 1960, “the estimated salary [of a teacher] was US$7,236 and 8.2 times per capita GDP in Korea, while it was $US4,869 and 4.5 times GDP per capita for the overall group of developing countries. This high teacher salary may indicate that Korean teachers were relatively well-qualified” (Lee J.-W., 2007, p. 7). However, this could also be the reflection of a cultural feature of the Korean society, in which teachers occupy a high social position.
Despite the existence of these possible explanations of how did Korea manage to reach such high levels of human capital accumulation, there is a possibility that has been underexplored by researchers, but that may explain the relatively high effectiveness of public and private investment in Korea. This could also explain part of the low educational outcomes displayed by Latin American countries both of average and top students: student effort.
High student effort and competition in Korea has been widely reported by the media, and has been present in the political debate for many years, including criticism of the College Scholastic Ability Test, the national examination for entry in the higher education system, and its apparent links to suicide in students (Card, 2005). Many believe that high effort levels might be explained by cultural traits and ethics. However, it is possible that some of the structural characteristics of the education system, such as egalitarian distribution of school quality, have created an environment that stimulates high levels of student competitiveness and effort directly or indirectly, and thus it could be explained endogenously by the [economic] decisions of students. That being said, a theoretical model can be capable of describing the role of culture and
structural elements of the education system on student effort, and consequently, on the efficiency and effectiveness of the education system.
2.4. Economic research on student effort, and the relation of education and social mobility
There is mixed economic evidence of the impact of student effort on academic achievements and performance, although most research is aligned with common sense, and supports the hypothesis that the impact is positive and statistically significant. On the one hand, there are researchers such as Prince, Kipps, Wilhelm and Wetzel (1981) show that effort contributes significantly to scholastic achievement. In other words, student effort is positvely related to students' academic performance. Similarly, Rich (2006) found that students' efforts have a significant positive impact on their performance, using observed rather than self reported measures of effort. In specific terms, Rich observed that students have a better performance in their exams if they made an effort to: i) do their homework, ii) attend to class, iii) arrive on time and iv) participate in class (2006, pp. 131-133).
For other scholars, student effort is not only significantly related to performance but it is the most relevant and statistically significant factor in determining performance (Trejos & Barboza, 2008, p. 29). These researchers used a sample of students from maroeconomics and statistics classes, and tested the effect of possible factors on academic performance measured by overall grade and final test scores. They conclude that effort measured as the cummulative measure of student’s dedication and involvement to the learning process is the most important factor, above measures of predetermined performance and ex-ante abilities such as GPA and SAT scores (Trejos & Barboza, 2008). Along with this view, De Fraja, Oliveira, & Zanchi (2012) developed and tested a model in which effort from students, parents and schools are simultaneously determined as a Nash equilibrium. They tested their model empirically and concluded that effort affects examinations and other outcome measures directly. Their work suggests that affecting effort directly has an impact on educational results, and that there are synergies and complementarities among these three types of effort involved.
On the other hand, there are researchers such as Fryer (2011) whose work seem to provide evidence that the students lack knowledge about the education production function, that is, they have problems translating effort into academic achievement. These articles focus on the
effectiveness of providing students with economic incentives. Their results might be ultimately interpreted as i) student effort is not significant as a determinant of educational outcomes due to inability of the students to translate effort into achievement; ii) complementarities in the education production function, for example good teachers and student effort might not be accounted for; iii) unpredictability of outputs by noisy correlation of school inputs and effort to outcomes (large error terms) might eliminate the incentives to exert effort or; iv) these incentives might replace intrinsic motivation with extrinsic motivation, and as a consequence attenuate or completely annul its effect on student effort11. Most probably, a combination of these interpretations is most accurate.
Other researchers have focused on student effort indirectly. Student effort could be theoretically used as a measure of the link between between cronyism and corruption on the labor market to educational attainment as measured by PISA test scores. These authors carry out an econometric exercise to determine whether there is a statistically significant link between these two variables. They find the relation and interpret ‘incentive effects’ over student effort as the mechanism through which cronyism and corruption may negatively affect educational attainment (Coco & Lagravinese, 2012).
Additionally, there has been some research, on how different school phenomena affect academic outcomes through student effort. Akerlof and Kranton (2002) have studied how social dynamics in schools and the identity problem in students (widely recognized sociological issues) affect both expenditure efficiency and overall academic outcomes through their impact on student effort. By including the different social identities (such as the nerds, the athletes, the burnouts and others) in the student’s utility functions, they were able to study their impact on student effort, and consequently, on academic performance. Their conclusions lead us to believe that student effort is a relevant analytical category that may allow us to capture phenomena and draw conclusions that are not directly accessible by conventional economic analysis.
Another relevant case worth noting is the research carried out by Ayşegül Şahin (2004), who studied the effects of higher education subsidies on overall academic performance. He constructed a highly sophisticated game theory model, which enabled him to link higher education subsidies to lower academic performance through the effects of lower pressure of parents (who have more incentives to exert pressure on their children if fees are higher) to keep
effort levels high.
On the other hand, there is also research of the impact of the education system on social mobility and economic inequality. Perhaps the Becker-Tomes model (1986), has been the most influential theoretical model on this subject, and can be considered the paradigm of the relationship between economic inequality, and social mobility through the education system (Gaviria, 1998, p. 4). This model focuses on altruistic parents as the decision-making unit, and expenditure on education as their choice variable. The degree of intergenerational mobility is determined by the interaction of their behavior on investment and consumption with the effects of borrowing constraints, and number of children in the household. The authors examine a number of empirical studies for different countries. Almost all the earnings advantages or disadvantages of ancestors are wiped out in three generations. Although this model is very powerful and its conclusions are very interesting, it does not allow us to analyze the implications of education policies or strategies such as the ones pursued by Korea and the effects of the implementation of the egalitarian principle on their education system. In this sense, the model proposed in the next section could be a valuable complement for this analytical exercise. The simultaneous inclusion of policy-dependent disparities in school qualities and student effort as an endogenous variable has not been attempted before, and could generate interesting findings on efficiency and growth implications, complementing its social mobility and inequality implications, thus covering the two main drivers of poverty reduction.
3. Construction of the basic analytical model
The purpose of this chapter is to build the simplest theoretical framework that can describe the mechanisms through which inequality in the education system can affect student effort, based on what have been presented in the previous sections. The aim of this exercise is to visibilize the effects of inequality in the education system on the two drivers of poverty reduction: economic growth (through human capital accumulation) and a simple measure of intergenerational social mobility.
The objective is to model the mechanism through which social inequality in the education system affects the level of effort exerted by students, so we must first state the basic assumptions underlying the education system and student behavior. We will focus on inequality as differences in the school inputs that are available for students from different social classes. These inputs are defined in the broadest sense in order to cover all the school-level variables that have been recognized to affect educational outcomes such as teacher quality (both measured as their professional qualifications and their pedagogical abilities), physical and technological infrastructure, total class time per day, among others. As this framework has been designed to be as simple as possible, it will assume only 2 students (one from a higher social class and one from a lower social class), and that the education system only offers coverage for one of them in the higher education system. Both of them are supposed to share the same ‘innate abilities’. It is also assumed for simplicity that there are only two different education levels: finished higher education and finished secondary education (high school), each one of them allowing the student to receive a different wage in the labor market. The place in higher education will be allocated to the student that scores higher in a state exam, which is specifically designed to measure the valuable marketable knowledge and abilities accumulated by students. This basic framework has the advantage that it allows this situation to be described by the “power” or “ratio” form of the conflict technology specification in economic conflict theory (Jia & Skaperdas, 2011). However, as this type of treatment is only able to account for inter class competition neglecting intra class competition, it cannot be used to analyze important issues such as higher education coverage, as well as changes or differences in the underlying static social structure.
Additionally, we will make what is probably the strongest assumption in this model: that the only difference between social classes is the quality and quantity of inputs of the school they are able to access. It will also be assumed that every combination of quantity and quality of school inputs constitute a total ordering according to their partial contribution to the attainment of educational outcomes. We could call this, its ‘pedagogical contribution’. This will allow us to assign a real (positive) value to every such combination. Although there are considerable theoretical limitations of imposing such a restrictive assumption, it is important for two reasons. The first one is that it allows us to keep the model in a simple form, while describing the basic mechanism through which school input inequality affects student effort. Second, if we were able to demonstrate these effects, even in the absence of other asymmetries generated by social class,
in important aspects such as cultural endowments, information asymmetries, impatience levels and access to financial services, we would have still learned important lessons. These additional asymmetries could be included and analyzed in extensions of this basic model.
Furthermore, it will be assumed that the level of valuable marketable knowledge and abilities accumulated by students is an increasing function of both individual effort and the ‘pedagogical contribution’ of the particular combination of school inputs. And finally, we have to analyze the direct impact of effort in the student’s utility function. If we define effort as the allocation of resources (such as spare time and mental concentration) that are on one hand required to accumulate valuable abilities, but on the other hand, used to generate utility directly (through activities such as leisure and family), we conclude that utility is a decreasing function of individual effort, which describes its opportunity cost. These elements are enough to describe the tradeoff present in this model: recognizing i) that higher education attainment levels are related through various channels to higher incomes, ii) that higher effort increases the probability of accessing higher education, and iii) that effort is inversely related to leisure, the tradeoff described is a traditional one: consumption vs. leisure. The way in which school input inequality affects this tradeoff will determine its impact on equilibrium effort levels, which is the main objective.
3.2. Description of the core problem
According to the assumptions and descriptions in the previous section, the core problem is the maximization of the expected utility function of the student, which is defined as an increasing function of his expected future permanent income, and a decreasing function of the effort that he decides to exert on his education. In order to build the expected utility function, it is important to start by the utility function, which is assumed by simplicity to be a regular Cobb-Douglas function:
(1) 𝑈 =𝑥!𝑙!!!
Where: x represents consumption of private goods and l stands for leisure. Decreasing marginal utility on both implies 𝛽 ∈ 0,1 . Normalizing prices to 1, we have 𝑥 =𝑤!, where w is
the wage level that can be high or low, depending on the education level (higher or secondary education), that is 𝑘 = ℎ,𝑙 .
Since effort has been defined as the allocation of resources such as spare time and concentration to studying, if the total amount of these resources were normalized to 1, it can be described as: 𝜀+𝑙 =1, which implies 𝑙= 1−𝜀. In this context, 𝜀∈ 0,1 , and represents the
proportion of these resources that is allocated to studying.
As a result the individual utility function can be rewritten in terms of wages and efforts for student i:
(2) 𝑈! =𝑤!!! 1−𝜀
! !!!
By applying the natural logarithm, it is possible to obtain another expression of the utility function that represents the same preferences:
(3) 𝑈!"
! = 𝛽𝑙𝑛 𝑤!! + 1−𝛽 𝑙𝑛 1−𝜀!
However, since the wage level is unknown when the student decides to exert effort on his studies, the student maximizes a Von Neumann-Morgenstern expected utility function:
(4) 𝑈!
! = 𝑝 𝛽𝑙𝑛 𝑤!! + 1−𝛽 𝑙𝑛 1−𝜀! + 1−𝑝 𝛽𝑙𝑛 𝑤!! + 1−𝛽 𝑙𝑛 1−𝜀! Where:
p is the probability that student i wins, that is, he outperforms his competitor in the state exam, and gets the only place in the higher education institution.
The form of p is based on the simplest form used in conflict and contest models, the “power” or “ratio” form, where the power is 1:
(5) 𝑝= !!
!! !
!!! =
ℎ! ℎ!!ℎ!=
ℎ! ℎ!
!! ℎ! ℎ!
Where h is a measure of the valuable abilities that have been accumulated by students through their activities in the education system. Because of imperfect correlation between student ability and the grade in state exams (that is, the existence of a noisy ‘error’ term), it is reasonable to interpret p as the probability that student 1 attains a higher grade in the state exam than student 2, thus obtaining the space in higher education. This means that a higher ability does not necessarily ensures the space in higher education, but it increases the probability. This also means that for larger average error terms (for example, for higher noise in the determination of the definitive grade in the state examination), the probability function would be less sensitive to changes in the levels of individual abilities. It is reasonable to note, ex-ante, that this would affect the students’ incentives to exert effort, as the link between effort and a desired outcome is reduced.
This specific form has the following desirable characteristics for a conflict technology specification, as ∀ℎ! ∈ℝ!∖ 0 :
𝑝∈ 0,1 That is, probabilities will not be negative nor greater than unity.
𝜕!
𝜕ℎ!> 0 That is, probability is increasing on abilities.
!!
!!! <0 That is, probability is decreasing on the competitor’s abilities.
Despite the technical advantages of this simple technological specification, we should point out two important limitations. First, we can easily realize that the probability described by this form seems too insensitive to differences in abilities. For example, in the case in which student 1 has twice the abilities of student 2, a situation in which one may intuitively conclude that the probability that student 1 is able to outperform student 2 in the state exam is very high (p very close to 1), the resulting probability is a mere p = 2/3, which seems unrealistically low. The second limitation is a consequence of the “ratio” characteristic of the form (which means that the ratio of abilities is the sole determinant of the probabilities). Since this is so, there is a very strong
incentive to avoid 0 and near 0 ability levels, as the marginal effect of changes in ℎ! on ℎ!
ℎ! when ℎ! = 0 approaches an infinite value. In other words, this rules out the possibility of total deterrence. Nevertheless, we will show that these limitations will not hinder the model in the attainment of its main results. Still, the possibility of extending the model to more sensitive conflict technologies and allowing for total deterrence could yield interesting results.
Since the accumulation of abilities is assumed to depend only on school inputs and and the level of individual effort, a simple functional form that allows for constant returns on quality and diminishing returns for effort is assumed:
(6) ℎ! = 𝑞!𝜀!!
Where 𝑞! is the positive number representing the ‘pedagogical contribution’ of the particular combination of school inputs available to student i. This functional form is consistent with these two elements exhibiting both complementary (mutually reinforcing – more and better school inputs can intensify the impact of effort on ability accumulation and vice versa) and
substitute properties (mutually compensating – the impact of lower ‘pedagogical contribution’ of school inputs on the accumulation of abilities can be compensated with greater student effort).
3.3. Maximization and results
The maximization of the expected utility of student 1 by choosing the optimal effort level is described by the following expression:
7 max
!! 𝑝 𝛽𝑙𝑛 𝑤
! + 1−𝛽 𝑙𝑛 1−𝜀
! + 1−𝑝 𝛽𝑙𝑛 𝑤! + 1−𝛽 𝑙𝑛 1−𝜀!
From which we obtain the following first order condition:
(8) 𝛽𝜕!𝜕!
! ln 𝑤
! −ln 𝑤! = !!! !!!!
On the left hand side we have the marginal utility of effort through the probability of accessing higher education (and higher future incomes), and on the right hand side we have the marginal cost of effort through reduced leisure.
Replacing (6) for both students in (5), we obtain an expression of the probability in terms of individual efforts and high school qualities:
(9) 𝑝= !!!!!
!!!!!!!!!!!
By differentiating that expression with respect to 𝜀!, we get 𝜕𝜕!!! = !"!!!
!!!! !!!!
!!!!!!!!!!! ! which we
then replace on the first order condition to obtain the optimality condition in terms of the parameters, efforts and high school qualities.
(10) 1−𝜀! !!!!!!!!!!!!! !!!!!!!!!!! ! =
!!! !
!
!" !! !!
Equation (10) is the implicit reaction curve of student 1. Since the problem is symmetrical (we have assumed that the only difference between the high class student and the low class student is the pedagogical contribution of their school inputs), one easily obtains the implicit reaction curve of student 2:
(11) 1−𝜀! !!!!!!!!!!!!! !!!!!!!!!!! ! =
!!!
! ! !" !!
!!
In order to obtain the equilibrium, and since the right sides of both equations are the same and consist solely con parameters, we can equalize both left sides and obtain:
1−𝜀! 𝛼𝑞!𝜀!!!!𝑞!𝜀!! 𝑞!𝜀!! +𝑞
!𝜀!! !
= 1−𝜀! 𝛼𝑞!𝜀!!!!𝑞!𝜀!!
𝑞!𝜀!! +𝑞 !𝜀!! ! Which is easily simplified to
1−𝜀! 𝑞!𝜀!!!!𝑞
!𝜀!! = 1−𝜀! 𝑞!𝜀!!!!𝑞!𝜀!!
And with further algebraic manipulation we arrive to the conclusion that in equilibrium:
(12) 𝜀! = 𝜀!
That is, as expected, the equilibrium is also symmetrical. This is the first result of this model and is very interesting despite its simplicity and triviality. Its economic interpretation is that differences in the qualities of both high schools will not generate differences in effort levels among students of different social strata, in the absence of other asymmetries generated by social status. We are aware that this is an unrealistic result, and is the consequence of oversimplification and assuming-out the asymmetries that exist between social classes, as stated at the beginning of this section. However, it is useful because it demonstrates that in order to explain observed differences in effort levels across social strata, we would have to recur to other asymmetries generated by social status such as credit constraints, access to private tutoring, cultural differences (for example, different valuations of leisure, future incomes, impatience, etc.); not school input inequality. It also conveys the important conclusion that both students respond to changes in the parameters of the model in the same direction.
We will now construct a basic measure of intergenerational social mobility. Assuming without loss of generality that 1 is the student from a more favorable socioeconomic background, that is, 𝑞! > 𝑞!, and that p is the probability that student 1 outperforms student 2 in the state examination, if we define:
Then, 𝑠∈ 0,1 , and it will be a measure of social mobility. As s approaches 1, both students will have the same probability of achieving the highest score on the state examination; and as s approaches 0, the probability gap that favors student 1 is wider. In other words, there is a higher correlation between the socioeconomic background and the future income level. Using (5)
in (13) we have: 𝑠= !!! ! = !! !!!!! !! !!!!! =!!
!!, which means that social mobility will be a measure of the
relative abilities of the students. Now, using equation (6), we have: 𝑠= !!!!! !
!!!!, however with
equation (12) we can simplify to 𝑠 =!!!!
!!!! and finally to:
(14) 𝑠 = !!
!!
That is, social mobility can be completely described as the quotient of the respective pedagogical contribution of school inputs, as a measure of school input inequality. This is an expected result, since it is common knowledge that social mobility is stimulated by leveling the playing field, by increasing equality of opportunity.
We will now use (12) in (10) to obtain the equilibrium level of effort for this education system. Since 𝜀! = 𝜀!, we will now use 𝜀= 𝜀! =𝜀!. Replacing, we obtain
1−𝜀 !!!!!!!!!!!! !!!!!!!! ! =
!!! !
!
!"!! !!
, which can be simplified first to:
1−𝜀 !!!!!!!!!!!!!!
!!!! !=
!!!
!
! !"!!
!!
and finally to: !!!! = !!"!! ! !"!!
!!
!!!!! !
!!!! .
And so, solving for 𝜀 we obtain the second –and most important result– of this model:
(15) 𝜀∗ = !
!! !!!"! ! !"!!
!!
!!!!!! !!!!
3.4. Interpretation and implications of the main result
The different interpretations that one may derive from this equation are:
•The level of effort depends negatively on the fraction !!"!! , which means it is directly
as well as to the effect of personal effort on ability accumulation (𝛼). This result is expected and consistent with intuition.
•The level of effort depends positively on ln!!!!, which means that the larger the ratio
between the expected incomes (larger return on higher education) the higher the equilibrium effort level will be. This is also expected and consistent with intuition.
•The level of effort depends negatively on the fraction:
(16) !!!!! !
!!!!
This last one is the most difficult to interpret, but is our most important result, as we will prove that this fraction is only dependent on the inequality (fraction) between school inputs, not on their absolute level.
If we define the following:
(17) 𝑎= 𝑞!
(18) 𝛾𝑎 =𝑞!
we can rewrite expression (16) as !!!!!"! !, which can be simplified to:
(19) !!!!! !
!!!! = !!! !
!
Thus, the value of the fraction !!!!! !
!!!! can be completely described as a function of 𝛾.
Next, we will show that the parameter 𝛾 is a non-linear measurement of school input inequality:
Assuming without loss of generality that 1 is the high-class student and that 2 is the low-class student, that is, 𝑞! ≥𝑞! and consequently 𝛾 ∈ 0,1 , we use (17) in (18) and obtain
(20) 𝛾𝑞! =𝑞!
which is simplified to:
(21) 𝛾 = !!
!!
Equation (21) allows us to interpret 𝛾 as the quotient of 𝑞!and 𝑞!, and by recalling equation (14), as a measure of social mobility.
