CHAPTER 13 Education production functions Eric A. Hanushek Hoover Institution, Stanford University, CESifo, IZA and NBER, Stanford, CA, United States

Glossary The attention to education production func- tions is driven largely by recognition that indi- Educational production function A function that relates various inputs to education including those of families, vidual skills have significant payoffs in the peers, and schools to the maximum level of student labor market and elsewhere. Thus, a natural achievement that can be obtained question is how skills can be developed and Fixed effects A form of statistical analysis that removes the enhanced, leading to the analysis of how schools average effects of a factor (such as individual schools) and other educational inputs enter into skill from the analysis; in the case of teachers, the fixed effect in models of achievement growth is often interpreted as a development. measure of teacher quality This discussion focuses largely on evidence Value-added In the context of education production func- from the US where there is a lengthy history of tions, the value-added of an input would be the separate analysis of education production functions. The contribution of learning after allowing for other inputs focus of this work has changed over time, mov- and the base level of knowledge of the students ing from standard inputs and resources to the effectiveness of teachers. This analysis has been Overview aided by the development of much more exten- sive data bases on school performance that Much of the analysis in the economics of edu- come from school accountability systems. There cation flows from a simple model of production. has also been wider analysis of educational pro- The common inputs are things like school re- duction from other countries (Woessmann, sources, teacher quality, and family attributes; 2016). and the outcome is some measure of student achievementdfrequently but not always student test scores. Knowledge of the production func- Measuring skills and human capital tion for schools can be used to assess policy alter- natives and to judge the effectiveness and Education production functions have their efficiency of public provided services. This area roots in the more general analysis of human cap- of research is, however, distinguished from ital, but the two different streams of analysis many because the results of analyses enter quite have largely diverged in the past. Most human directly into the policy process. capital analysis has strong and direct linkages

The Economics of Education, Second Edition https://doi.org/10.1016/B978-0-12-815391-8.00013-6 161 Copyright © 2020 Elsevier Ltd. All rights reserved. 162 13. Education production functions to labor market outcomes and the determination happens in schoolsdthus, it does not provide a of earnings. Education production functions, complete or accurate picture of outcomes. It as- while ostensibly closely related, have focused sumes a year of schooling produces the same more on the underlying determination of skills amount of student achievement, or skills, over and human capital. Until recently, the two time and in every country. Importantly, it also different foci have led to quite different perspec- assumes that schooling is the only input into skill tives on both the measurement of human capital development, ignoring the extensive contrary and implicitly the fundamental modeling of eco- evidence discussed below. Finally, this measure nomic outcomes. of school outcomes ignores the extensive policy Historically, human capital has been consid- debates about ways to improve school quality. ered from the labor market perspective of the in- A common extension in the investigation of dividual. In its simplest form, individuals make individual human capital and labor market out- investments that develop their skills, and this comes is the addition of cognitive skills as stock of skills is optimized for the labor market. gauged by standardized test scores to the empir- In this analysis, the most frequently employed ical models. Such skill measures have found measure of individual skills, or human capital, their way into the literature slowly, because has been school attainment, or simply years of they have just been available for a limited num- schooling completed. ber of surveys that include both achievement There are several justifications for relying on and labor market outcomes. Moreover, in these years of schooling to measure individual skills. analyses the common interpretation of these First, a prime motivation for the schools is the measures is that they represent individual “abil- acquisition of knowledge and skills, and this jus- ity” and thus can be added to a standard Mincer tifies the heavy governmental investment in earnings function to correct for any school selec- schools of nations around the world. Second, in tion bias arising from people with higher ability the early development of human capital theory, continuing longer in school. Such investigations Mincer (1970, 1974) developed a simple but of the impact of cognitive achievement include, elegant investment model for individuals that for example, Lazear (2003); Mulligan (1999); emphasized time spent in school. This theoretical and Murnane, Willett, Duhaldeborde, & Tyler development translated into one of the most suc- (2000)). In these, however, the general interpreta- cessful empirical modelsdthe “Mincer earnings tion is still that school attainment is the measure function”dthat relates individual earnings to of human capital. school attainment and to labor market experi- A complementary line of research has consid- ence. The value of school attainment as a rough ered aggregate human capital and how it affects measure of individual skill has subsequently national productivity and growth. The surge in been verified by a wide variety of studies of la- empirical analyses of growth differences across bor market outcomes (Card, 2001). Third, reli- countries begun in the early 1990s invariably ance on years of school as the human capital included measures of school attainment to reflect measure is expedient. School attainment is very the skills of the population (see Barro, 1991; commonly measured in censuses and surveys, Romer, 1990). In a subsequent comparison of allowing, for example, estimation of Mincer alternative drivers of growth rate differences, earnings functions in 139 countries (see the re- Sala-i-Martin, Doppelhofer, and Miller (2004) view in Psacharopoulos & Patrinos, 2018). found primary school enrollment to be among However, the difficulty with this common the strongest explanatory factors. Nonetheless, measure of outcomes is that it simply counts the estimated impacts of human capital and the time spent in schools without judging what other inputs appeared very unstable, casting

III. Production, costs and financing Basic production function estimates 163 doubt on the line of empirical growth analyses readily interpreted in a human capital frame- (Levine & Renelt, 1992). work. Similarly, looking at long run growth in The parallel line of research into education termsofcognitiveskillshelpsresolvemanyof production functions has concentrated on under- the difficulties with empirical growth models standing the determinants of human capital. Test built on school attainment (Hanushek & Woess- scores, or measures of cognitive skills more mann, 2008). Hanushek and Kimko (2000) generally, have been interpreted as proxies for demonstrate that quality differences measured skills that are valued in the labor market and by achievement have a dramatic impact on pro- elsewhere and, as such, more immediate mea- ductivity and national growth rates. This is rein- sures of human capital differences. As described forced and extended in Hanushek and below, the overall focus of this work has then Woessmann (2012, 2015), who show that been understanding how schools and other fac- three-quarters of the variation in country tors determine the skills of individuals. growth rates can be explained by a simple But, this latter focus also leads to a very growth model that focuses on cognitive skills. different interpretation of the prior labor market Theycalltheaggregatetestscoresforcountries studies focused on school attainment. Moreover, “knowledge capital” to distinguish it from the reconsideration of the interpretation of school attainment measures that are frequently measured cognitive skills in the prior human referred to as being synonymous with human capital/labor market analyses helps to clarify capital. the problems with these prior analyses and to These recent analyses of labor market rela- point toward an alternative empirical approach. tionships with standardized achievement test Specifically, these achievement measures have scores complete the linkage between human cap- been interpreted as proxies for skills that are ital analysis from the individual labor market valued in the labor market, albeit the ability perspective and education production functions designation implies that these are fixed skills. that seek to explain differences in test scores. An alternative interpretation is that the tested This linkage provides the rationale for interpret- skills represent an explicit measure of human ing the results of education production estimates capital. If so, whether skills were determined as indicating the longer run economic effects of by schools or by other inputs would in general schools and other inputs. not affect their use and interpretation in under- standing variations in labor market outcomes. Thinking of skills and their measurement by Basic production function estimates test scores in this manner implies that school attainment is just one of a variety of inputs into Analysis of education production functions an individual’s skills. has a direct motivation. Because educational out- The research considering cognitive skills as a comes cannot be changed by fiat, much attention direct measure of human capital goes a long has been directed at inputsdparticularly those way toward resolving some of the apparent perceived to be relevant for policy such as school anomalies in the prior labor market research. resources or aspects of teachers. Using data for a representative sample of 23 Analysis of the role of school resources in countries that included test scores along with la- determining achievement begins with the Cole- bor market information on individuals, man Report, the US government’s monumental Hanushek, Schwerdt, Wiederhold and Woess- study on educational opportunity released in mann (2015) show that estimates of earnings 1966 (Coleman et al., 1966). While controversial, functions in terms of achievement tests are that study’s greatest contribution was directing

III. Production, costs and financing 164 13. Education production functions attention to the distribution of student Measured school inputs performancedthe outputs as opposed to various school inputs such as spending per pupil or char- The central thrust of education production acteristics of teachers (Bowles & Levin,1968; function estimation has changed over time. Dur- Hanushek & Kain, 1972; Hanushek, 2016). ing the initial perioddroughly 30 years starting The underlying model that has evolved as a with the Coleman Reportdthe analyses fol- result of this research is very straightforward. lowed a common pattern of examining the The output of the educational processdthe impact on student learning of specific measures achievement of individual studentsdis directly of school inputs. These studies focused on related to inputs that both are directly controlled measured resources of the type typically by policymakers (for example, the characteristics included in school reports. The second period, of schools, teachers, and curricula) and are not so following this initial estimation phase and car- controlled (such as families and friends and the rying through to the present, moved to an exam- innate endowments or learning capacities of ination of specific aspects of production, often the students). Further, while achievement may with novel methods or data, and to concentra- be measured at discrete points in time, the tion on teacher effects. educational process is cumulative; inputs The state of knowledge about the impacts of applied sometime in the past affect students’ cur- basic resources is best summarized by review- rent levels of achievement. ing available empirical studies from the first Family background is usually characterized period. Most analyses of education production by such socio-demographic characteristics as functions directed their attention at a relatively parental education, income, and family size. small set of resource measures, and this makes Peer inputs, when included, are typically ag- it easy to summarize the results (Hanushek, gregates of student socio-demographic charac- 2003). The 90 individual publications that teristics or achievement for a school or appeared before 1995 contain 377 separate pro- classroom. School inputs typically include duction function estimates. For classroom re- teacher background (education level, experi- sources, only nine per cent of estimates for ence, sex, race, and so forth), school organiza- teacher education and 14% for teacherepupil tion (class sizes, facilities, administrative ratios yielded a positive and statistically signif- expenditures, and so forth), and district or icant relationship between these factors and community factors (for example, average student performance. Moreover, these studies expenditure levels). Except for the original were offset by another set of studies that found Coleman Report, most empirical work has a similarly negative correlation between those relied on data constructed for other purposes, inputs and student achievement. Twenty-nine such as a school’s standard administrative re- per cent of the studies found a positive correla- cords. More recent work has moved to use tion between teacher experience and student micro-data generated from school account- performance; however, 71 percent still pro- ability programs. Statistical analysis (typically vided no support for increasing teacher experi- some form of regression analysis) is employed ence (being either negative or statistically to infer what specifically determines achieve- insignificant). Subsequent analysis of experi- ment and what is the importance of the various ence effects consistently indicates that increased inputs into student performance. Over time, experience for the first few years of teaching has attention has been increasingly directed at the a positive impact, but there is little to no addi- statistical identification of factors that are caus- tional impact past the initial teaching period ally related to student outcomes. (see Hanushek & Rivkin, 2012).

III. Production, costs and financing Basic production function estimates 165 Studies on the direct effect of financial re- the certification and hiring of teachers, and the sources provide a similar picture, although like. If these attributes are importantdas much here the analysis has been more controversial policy debate would suggestdthey must be (see, for example, Hanushek,1994, 2003; Hedges, incorporated into any analysis of performance. Laine, & Greenwald, 1994). These indicate that The adequacy of dealing with these problems there is very weak support for the notion that can thus be used as a simple index of study simply providing higher teacher salaries or quality. greater overall spending will lead to improved The details of these quality issues and ap- student performance. Per pupil expenditure has proaches for dealing with them are discussed received the most attention, but only 27 percent in detail elsewhere (Hanushek, 2003) and only of studies showed a positive and significant ef- summarized here. The first problem is amelio- fect. In fact, seven per cent even suggested that rated if one uses the “value added” versus adding resources would harm student achieve- “level” form in estimation. That is, if the achieve- ment. It is also important to note that studies ment relationship holds across grades, it is involving pupil spending have tended to be the possible to concentrate on the growth in achieve- lowest-quality studies as defined below, and ment and on exactly what happens education- thus there is substantial reason to believe that ally between those points when outcomes are even the 27 percent figure overstates the true ef- measured. This approach ameliorates problems fect of added expenditure. of omitting prior inputs of schools and families, These studies make a clear case that resource because they will be incorporated in the initial usage in schools is subject to considerable ineffi- achievement levels that are measured ciency, because schools systematically pay for in- (Hanushek, 1979). The latter problem of impre- puts that are not consistently related to outputs. cise measurement of the policy environment These results of course do not indicate that can frequently be ameliorated by studying per- money never matters or that money cannot mat- formance of schools operating within a consis- ter. They instead point to the importance of how tent set of policiesdfor example, within money is spent rather than how much is spentda individual states in the US. Because all schools topic considered below. within a state operate within the same basic pol- icy environment, comparisons of their perfor- mance are not strongly affected by unmeasured policies (Hanushek, Rivkin, & Taylor, 1996). Study quality If the available studies are classified by The previous discussions do not distinguish whether or not they deal with these major qual- among studies on the basis of any quality differ- ity issues, the prior conclusions about research ences. The available estimates can be reasonably usage are unchanged (Hanushek, 2003). The categorized by a few objective components of best quality studies indicate no consistent rela- quality. First, while education is cumulative, tionship between resources and student out- frequently only current input measures are avail- comes. The studies finding strong resource able, which results in analytical errors and effects, particularly for expenditure per pupil, biased estimates of the effects of specific inputs. are heavily concentrated in the group of lowest Second, schools operate within a policy environ- quality studies. ment determined almost always by higher levels An additional issue, which is particularly of government. In the United States, state gov- important for policy purposes, concerns whether ernments establish curricula, provide sources of this analytical approach accurately assesses the funding, govern labor laws, determine rules for causal relationship between resources and

III. Production, costs and financing 166 13. Education production functions performance. If, for example, school decision- outcomes can be observed. Importantly, such makers provide more resources to those they studies also can validate general estimates of judge as most needy, higher resources could sim- school and teacher effects by, for example, ply signal students known for having lower showing the direct linkages of estimated school achievement. Ways of dealing with this include factors on subsequent earnings for the students various regression discontinuity or panel data (Chetty, Friedman, & Rockoff, 2014). approaches. When done in the case of class sizes, There has also been continuing study of the ef- the evidence has been mixed (Angrist & Lavy, fects of simply adding more funding on outputs. 1999; Hoxby, 2000; Rivkin, Hanushek, & Kain, This work has in part been motivated by the 2005). introduction of production function estimates into court cases about the financing of schools (Hanushek & Lindseth, 2009). In fact, two recent More recent studies studies use the imposition of court judgments The most significant innovation of recent against state funding rules to obtain estimates years is the use of large administrative data ba- of the impact of added funding (Jackson, John- ses. These data bases employ state or local re- son, & Persico, 2015; Lafortune, Rothstein, & cords on individual student’s performance and Whitmore Schanzenbach, 2018). The estimation are most notable for tracking students across and interpretation of these is the subject of on- grades. Student performance is then related to going research. that programs and personnel that each student A final alternative involves the use of random is exposed to over time. These large scale data- assignment experimentation rather than statisti- bases, often following all students in a state cal analysis to break the influence of sample se- over time, permit controlling for a wide range lection and other possible omitted factors. With of influences on achievement through the intro- one major US exception, this approach nonethe- duction of fixed effects for schools, individuals, less has not been applied to understand the and time (see, for example, Rivkin et al., 2005 impact of schools on student performance. (Ran- or Boyd, Grossman, Lankford, Loeb, & Wyckoff, domized trials have expanded much more 2006). These fixed effects hold constant any sys- rapidly in developing countries; see Duflo, Glen- tematic differences that do not vary within a nerster, & Kremer, 2007). The US exception is category (such as constant differences among Project STAR, an experimental reduction in class the sampled schools in terms of the selection of sizes that was conducted in the US state of Ten- schools by families and teachers) and obtain esti- nessee in the mid-1980s (Word et al., 1990). To mates of various inputs from their variation date, the use of randomized experiments has within each of the schools. By eliminating sys- not had much impact on research or our state tematic selection and sorting of students and of knowledge about the impacts of resources. school personnel, they can concentrate on spe- While Project STAR has entered into a number cific causal factors that determine individual stu- of policy debates, the interpretation of the results dent outcomes. remains controversial because of concerns about An additional aspect of the growing state data the quality of the experiment (Krueger, 1999; bases is that students can now be traced to sub- Hanushek, 1999). The results of this experiment sequent outcomesduniversity attendance, labor suggested a significant but small impact of lower market experiences, criminal behavior, and class size but that all of the impact was concen- more. This type of study, while just becoming trated in the first year of schooling (kindergarten more possible, involves linking data across state or grade one). Smaller class sizes in later years and federal programs where outside-of-school had no additional impact on student outcomes.

III. Production, costs and financing Do teachers and schools matter? 167 Do teachers and schools matter? equivalents during a school year, a good teacher gets gains of 1.5 grade level equivalents. Clearly, Because of the Coleman Report and subse- with a string of good or bad teachers, the impli- quent studies discussed above, many have cations for student performance could be very argued that schools do not matter and that large. only families and peers affect performance. Un- More modern research into state administra- fortunately, these interpretations have confused tive data bases have helped to refine the under- measurability with true effects. standing of the importance of differences in Extensive research since the Coleman Report teacher quality. For example, Rivkin et al. has made it clear that teachers do indeed matter (2005) are able to provide rough bounds on the when assessed in terms of student performance variation in teacher quality as seen within Texas instead of the more typical input measures based (the source of their administrative database). By on characteristics of the teacher and school. The these studies, one standard deviation in teacher alternative approach to the examination of quality implies around a 0.15 standard deviation teacher quality concentrates on pure outcome- in the growth of student achievement. By this, based measures of teacher effectiveness. The having a series of good teachers (teachers at the general idea is to investigate the “value-added 84 percentile of the quality distribution) instead of teacher” by looking at differences in growth of average teachers would lead to substantially rates of student achievement across teachers. A different learning after just a few years. For good teacher would be one who consistently ob- example, 4e5 years of a good teacher could close tained high learning growth from students, the average achievement gap between low in- while a poor teacher would be one who consis- come and higher income students. tently produced low learning growth. Early These estimates of magnitudes can be linked work relied upon very specialized samples of directly to studies of the economic impact of stu- students (e.g., Hanushek, 1971; Murnane, dent achievement. Hanushek (2011a, 2011b) 1975), but this has subsequently broadened out shows that a 75th percentile teacher each year considerably (Hanushek & Rivkin, 2010). generates over $400,000 in added income aggre- The general research design is to estimate gated over a class of 30 students (compared to an models of the growth in individual student average teacher). On the other hand, a 10th achievement that can be attributed to various percentile teacher subtracts $800,000 in aggre- measured school and family factors and to dif- gate from a class of 30 students (again compared ferences in learning across the students with to an average teacher). Using a very different different teachers (see reviews in Hanushek & estimation approach that links teacher value Rivkin, 2012; Jackson, Rockoff, & Staiger, 2014; added to the subsequent earnings of the specific Koedel, Mihaly, & Rockoff, 2015). The differ- students in the class, Chetty et al. (2014) confirm ences in student achievement growth across the order of magnitude of these teacher impacts. classrooms, which can be taken as a measure of These results can also be reconciled with the teacher quality, appear to be consistent and prior ones. These differences among teachers very large (Hanushek & Rivkin, 2010). are simply not closely correlated with commonly Hanushek (1992), for example, estimates that measured teacher characteristics (Hanushek, the variation in student outcomes from a good 1992; Rivkin et al., 2005). Moreover, teacher cre- to a bad teacher can be as much as a full year dentials and teacher training do not make a of knowledge per academic year; in other words, consistent difference when assessed against stu- while a poor teacher gets gains of 0.5 grade level dent achievement gains (Boyd et al., 2006;

III. Production, costs and financing 168 13. Education production functions Kane, Rockoff, & Staiger, 2006). Finally, teacher et al. (1990); Krueger (1999). Hoxby (2000) in quality does not appear to be closely related to her regression discontinuity estimation for Con- salaries or to market decisions. In particular, necticut schools finds no systematic effect of teachers exiting for other schools or for jobs class size. Rivkin et al. (2005), with their fixed ef- outside of teaching do not appear to be of higher fects estimation, find effects half of Project STAR quality than those who stay (Hanushek, Kain, in grade four and declining to insignificance by O’Brien, & Rivkin, 2005). grade seven. The analysis of teacher value-added demon- From a policy perspective the magnitude of strates the linkage between the research and alternative estimates is at best small. In order to policy discussions and decisions. Teacher be relevant for policy, it is necessary to compare value-added measures have been actively dis- the outcomes of any change with its costs. Most cussed in terms of teacher evaluations. In educational research ignores such comparisons reviewing US state policies, the National Council and neglects any consideration of costs. on Teacher Quality (2017) finds that by 2017 39 It is easy to see the importance of cost consid- states require teacher evaluations that include erations when put in the context of the debates objective measures of student achievement over class size reduction. In economic terms the growth, although the exact form and weight potential impacts of class size reduction are placed on these varies widely. very small when contrasted with the costs of This linkage with policy also heightens atten- such large class size reductions, which typically tion to the studies, and there has been extensive involve some of the most expensive policy analysis of the properties of value-added esti- changes currently contemplated. The relevant mates including their stability, their accuracy, alternative policy would be to compare the gains and their implications for the nature of teaching; from spending on class size reduction with the see, for example, Braun, Chudowsky and Koenig potential gains from improving the quality of (2010), Hanushek and Rivkin (2012), and Haertel teachers. (2013).

Some conclusions and implications Benefits and costs The existing research suggests inefficiency in Throughout most consideration of the impact the provision of schooling. It does not indicate of school resources, attention has focused almost that schools do not matter. Nor does it indicate exclusively on whether a factor has an effect on that money and resources never impact achieve- outcomes that is statistically different from ment. The accumulated research surrounding zero. Of course, any policy consideration would estimation of education production functions also consider both the magnitude of the impacts simply says there currently is no clear, system- and the costs of change. For magnitude of atic relationship between resources and student impact, even the most refined estimates of, say, outcomes. The general conclusion from the exist- class size impacts does not give very clear guid- ing work is that how resources are used is gener- ance. The experimental effects from Project ally more important than how much is used. At STAR indicate that average achievement from a the same time, more modern research into the reduction of eight students in a classroom would determinants of student achievement strongly increase by about 0.2 standard deviations, but indicates that teacher quality differences are the only in the first grade of attendance in smaller most significant part of differences across classes (kindergarten or first grade); see Word schools.

III. Production, costs and financing References 169 References Hanushek, E. A. (2016). What matters for achievement: Updating coleman on the influence of families and Angrist, J. D., & Lavy, V. (1999). Using Maimonides’ rule to schools. Education Next, 16(2), 22e30. estimate the effect of class size on scholastic Hanushek, E. A., Guido, S., Simon, W., & Woessmann, L. achievement. Quarterly Journal of Economics, 114(2), (2015). Returns to skills around the world: Evidence 533e575. from PIAAC. European Economic Review, 73, 103e130. Barro, R. J. (1991). Economic growth in a cross section of Hanushek, E. A., & Kain, J. F. (1972). On the value of equality countries. Quarterly Journal of Economics, 106(2), 407e443. of educational opportunity as a guide to public policy. In Bowles, S., & Levin, H. M. (1968). The determinants of scho- F. Mosteller, & D. P. Moynihan (Eds.), On equality of educa- lastic achievement–an appraisal of some recent evidence. tional opportunity (pp. 116e145). 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The knowledge cap- response to Hedges, laine, and Greenwald. Educational ital of nations: Education and the economics of growth. Cam- Researcher, 23(4), 5e8. bridge, MA: MIT Press. Hanushek, E. A. (1999). Some findings from an independent Hedges, L. V., Laine, R. D., & Greenwald, R. (1994). Does investigation of the Tennessee STAR experiment and money matter? A meta-analysis of studies of the effects from other investigations of class size effects. Educational of differential school inputs on student outcomes. Educa- Evaluation and Policy Analysis, 21(2), 143e163. tional Researcher, 23(3), 5e14. Hanushek, E. A. (2003). The failure of input-based schooling Hoxby, C. M. (2000). The effects of class size on student policies. Economic Journal, 113(485), F64eF98. achievement: New evidence from population variation. Hanushek, E. A. (2011a). The economic value of higher Quarterly Journal of Economics, 115(3), 1239e1285. teacher quality. Economics of Education Review, 30(3), Jackson, C. 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Jackson, C. K., Rockoff, J. E., & Staiger, D. O. (2014). Teacher Rivkin, S. G., Hanushek, E. A., & Kain, J. F. (2005). Teachers, effects and teacher related policies. Annual Review of schools, and academic achievement. Econometrica, 73(2), Economics, 6, 801e825. 417e458. Kane, T. J., Rockoff, J. E., & Staiger, D. O. (2008). What does Romer, P. (1990). Human capital and growth: Theory and certification tell us about teacher effectiveness? Evidence evidence. Carnegie-Rochester Conference Series On Public from New York city. Economics of Education Review, Policy, 32, 251e286. 27(6), 615e631. Sala-i-Martin, X., Doppelhofer, G., Ronald, I., & Miller. Koedel, C., Mihaly, K., & Rockoff, J. E. (2015). Value-added (2004). Determinants of long-term growth: A bayesian modeling: A review. Economics of Education Review, 47, averaging of cassical estimates (BACE) approach. The 180e195. American Economic Review, 94(4), 813e835. Krueger, A. B. (1999). Experimental estimates of education Woessmann, L. (2016). The importance of school systems: Ev- production functions. Quarterly Journal of Economics, idence from international differences in student 114(2), 497e532. achievement. The Journal of Economic Perspectives, 30(3), Lafortune, J., Rothstein, J., & Whitmore Schanzenbach, D. 3e32. (2018). School finance reform and the distribution of stu- Word, E., Johnston, J., Pate Bain, H., DeWayne Fulton, B., dent achievement. American Economic Journal: Applied Boyd Zaharies, J., Lintz, M. N., et al. (1990). Student/teacher Economics, 10(2), 1e26. achievement ratio (STAR), Tennessee’s K-3 class size study: Lazear, E. P. (2003). Teacher incentives. Swedish Economic Pol- Final summary report, 1985-1990. Nashville, TN: Tennessee icy Review, 10(3), 179e214. State Department of Education. Levine, R., & Renelt, D. (1992). A sensitivity analysis of cross- country growth regressions. The American Economic Review, 82(4), 942e963. Mincer, J. (1970). The distribution of labor incomes: A survey Further reading with special reference to the human capital approach. Ehrenberg, R. G., Brewer, D. J., Adam, G., & Douglas Journal of Economic Literature, 8(1), 1e26. Willms, J. (2001). Class size and student achievement. Psy- Mincer, J. (1974). Schooling, experience, and earnings. New chological Science in the Public Interest, 2(1), 1e30. York: NBER. Hanushek, E. A. (1986). The economics of schooling: Produc- Mulligan, C. B. (1999). Galton versus the human capital tion and efficiency in public schools. Journal of Economic approach to inheritance. Journal of Political Economy, Literature, 24(3), 1141e1177. 107(6, pt. 2), S184eS224. Hanushek, E. A. (2011). The economic value of higher teacher Murnane, R. J. (1975). Impact of school resources on the learning quality. Economics of Education Review, 30(3), 466e479. of inner city children. Cambridge, MA: Ballinger. Hanushek, E. A., & Woessmann, L. (2015). The knowledge cap- Murnane, R. J., Willett, J. B., Duhaldeborde, Y., & Tyler, J. H. ital of nations: Education and the economics of growth. Cam- (2000). How important are the cognitive skills of teen- bridge, MA: MIT Press. agers in predicting subsequent earnings? Journal of Policy Mishel, L., & Rothstein, R. (Eds.). (2002). The class size debate. Analysis and Management, 19(4), 547e568. Washington, DC: Economic Policy Institute. National Council on Teacher Quality. (2017). State teacher pol- Moe, T. M. (2011). Special interest: Teachers unions and Ameri- icy yearbook, 2017. Washington: National Council on ca’s public schools. Washington, DC: Brookings Institution Teacher Quality. Press. Psacharopoulos, G., & Patrinos, H. A. (2018). Returns to in- Peterson, P. E. (2010). Saving schools: From Horace Mann to vir- vestment in education: A decennial review of the global tual learning. Cambridge, MA: Belknap Press of Harvard literature. Education Economics, 26(5), 445e458. University Press.

III. Production, costs and financing