کمیت و کیفیت معلمان: دینامیک تجارت کردن

The paper addresses the two-fold rise in teacher–student ratio in the American K-12 school system in the post-World War II period accompanied by the evidence of a decline in the relative quality of teachers. We develop a dynamic general equilibrium framework for analyzing the teacher quantity–quality trade-off and offer an explanation to the observed trends. Our OLG model features two stages of education: basic and advanced (college), the latter required of teachers. The cost of hiring teachers is influenced by the outside opportunities for college graduates in the production sector. We show that the latter factor strengthens in the process of endogenous growth and that it affects the optimal trade-off between quantity and quality of teachers such that the number of teachers hired will grow over time while their relative, but not the absolute, human capital attainment will fall. This is accompanied by increasing inequality, among the group of college educated workers in particular. We show that this effect, which we call the rising talent premium, applies whether teacher salaries are determined based on merit pay or, alternatively, by collective bargaining. Moreover, the salary compression characterizing the latter regime exacerbates the loss of the more talented workers by the teaching profession. Further, we analyze a comparative dynamics effect of exogenous skill-biased technological change which raises the college premium. We show that the effect is detrimental to the aggregate quality of teachers and to the quality of basic education. An important insight from this analysis is that in the process of human capital driven economic growth the rise in premium for high ability outpaces that for the average, whereby this effect is accelerated by technological change. This puts a downward pressure on the “real” quality of education inputs and therefore can create a negative feedback effect on human capital development as a factor of economic growth.

A reduction of class sizes in K-12 schooling has been one of the main education policy priorities in the United States over the last several decades. This is evidenced by the dynamics of student–teacher ratio which fell from 25.8 in 1960 to 15.7 in 2005. Remarkably, this decline has persisted through the contemporaneous ups and downs of the enrollment dynamics (Digest of Education Statistics, 2007, table 61, reproduced in Table 1 below). Research, however, has shown that students' test scores have not risen despite increased individualized instruction. This has compelled policy makers and researchers to question the role of the quantity of teachers vs. their quality as factors in student performance (see Hanushek et al., 2005), particularly in light of a possible trade-off between the two.Indeed the changes in education statistics between 1955 and 2005 displayed in our Table 1 suggest the possibility of a quantity–quality trade-off in the supply of teachers over this period: a remarkable growth in the total number of teachers consistently outpacing the growth of student enrollments was accompanied by declining relative teacher salaries, despite essentially steady overall K-12 public education expenditures as a share of GDP since 1970. Another significant trend observed over about the same period is the decline in the aptitude of teachers relative to other educated workers. Hoxby and Leigh (2004) estimate that in 1963 41% of all teachers were of the “middle” aptitude relative to their educated peers, with 17% above and 42% below the average; by comparison, in 2000, 28% of all teachers were of the “middle” aptitude with 5% above and 67% below average. Corcoran et al. (2004) provide similar results. Thus the dynamics of the K-12 system exhibits a trade-off between the quantity of teachers and their quality. The presence of the trade-off between the two variables creates the well-known endogeneity challenge for the empirical efforts to estimate one of the variables while controlling for the other.2 These empirical results do not thus provide a conclusive basis to explain the observed policy bias in favor of the quantity of teachers (smaller classes) on the grounds of benefits to student performance. This underscores the need for a broader theoretical framework capturing the dynamic interaction between inputs in education as it is influenced by the labor market in the production economy. Our paper develops a theoretical framework for analyzing the teacher quantity–quality trade-off and offers an explanation to the observed trend biased in favor of quantity. We present a model which incorporates the factors of education quality discussed above, in a dynamic general equilibrium framework where education policy decisions affect and are affected by individual education and employment decisions, whereas the dynamics of human capital accumulation and labor productivity has a feedback effect on both. In our model, a public education agency wishes to maximize the quality of basic education per student and, due to a budget constraint, faces a trade-off between the quality and quantity of teachers to be hired. We explore the following factors that may affect this trade-off: (i) the opportunity cost of teachers in terms of potential earnings in the alternative occupations in the production economy, and its evolution in the process of economic growth; (ii) collective bargaining driven by teacher unions as an institutional factor leading to wage compression among teachers3; (iii) the rise of skill premium resulting from technological change, which affects the opportunity cost of teachers. In our model, basic education is publicly funded through an income tax, whose rate is set exogenously, and is administered by an education agency which allocates these funds while deciding on the levels of teacher salaries and the number of teachers to be hired. The model features two stages of education: basic and advanced (college), the latter being required of teachers. College graduates can also take jobs in the skilled labor force of the production sector and get paid a competitive wage according to their human capital attainment. This opportunity cost implies that the levels of teacher salaries set by the education agency will determine the quality (human capital levels) of teachers who can be hired. Thus the total cost of hiring teachers is affected in our model by the outside opportunities available to skilled individuals in the production sector.4 We show, moreover, that in the process of endogenous growth this effect strengthens and that it pushes the optimal trade-off between quantity and quality of teachers in the direction of the former. Namely, in the face of rising over time cost of highly able skilled workers, the education agency will find it optimal to opt for increasing the number of teachers hired while reducing the overall relative quality of the pool of teachers. (The absolute human capital attainment of teachers, however, will rise along with the overall human capital accumulation, while sliding toward the lower tail of the distribution of college educated population.) Furthermore, we show that this human capital dynamics is characterized by increasing inequality within the group of college educated workers as well as between it and the unskilled. In order to assess the role of an institutional factor in the teacher quantity–quality trade-off, we consider two alternative teacher employment regimes: the “merit pay” regime where each teacher's salary is based on his human capital attainment, as well as the “collective bargaining” regime where the education agency is bound by salary contracts with teacher unions which compress teacher salaries. Our results show that the salary compression due to the collective bargaining further exacerbates the loss of the more talented workers by the teaching profession and thus additionally contributes to the decline of teacher quality. We then build on these results to further analyze the impact of exogenous technological change biased toward skill, i.e., augmenting productivity of skilled workers and thereby the college premium. We show that such technological change will exacerbate the negative trends in the quality of basic education in relation to GDP growth. Specifically, the comparative dynamics effect (relative to the benchmark trajectory) will be detrimental to the aggregate quality of teachers as well as to the quality of basic education, due to the upward shocks to the cost of skilled labor. This is consistent with the aforementioned observation by Lakdawalla (2006) as well as the discussion by Rangazas (2002) of Baumol's Disease in education sector as a result of technological change driving up the sector's real labor costs, which leads him to conclude that school expenditures must be deflated by the rate of technological change in addition to the rate of inflation. Our model adds a dimension to this understanding: we show that in the process of human capital driven economic growth the rise in premium for high ability will outpace that for the average even besides the effect of technological change, hence an additional downward pressure on the “real” quality of education inputs. Thus this paper offers a theory explaining the trend in education policy in favor of lower student–teacher ratios (i.e., higher quantity of teachers) combined with deteriorating teacher quality, despite growing per student schooling expenditures. The paper is organized as follows. Section 2 develops a dynamic general equilibrium model of production economy and two-stage formation of human capital. In Section 3, we introduce the benchmark model of the administration of basic education with merit pay for teachers and then define and prove the existence of the recursive dynamic equilibrium. Section 4 extends the analysis to the collective bargaining regime for teachers; Section 5 delivers the main results on the dynamics of the quantity and quality of teachers for this case. Section 6 compares the teacher quantity–quality dynamics under the collective bargaining and merit pay regimes. Section 7 concludes. Appendix 1 presents a glossary of notation. Appendix 2 containing most of the proofs is posted at the Journal's website.

Over the last forty years, education policy in the U.S. has changed significantly, focusing in particular on lowering the student–teacher ratio. We have developed a model which offers an insight into this evolution by relating it to the changes in the US economy characterized by rising skill premium and overall income inequality. Our model predicts that as incomes of college educated individuals rise and become more dispersed (the phenomenon we call the “rising talent premium”), education policy-makers are forced to adjust relative teacher salaries and thereby the quality standards. Education quality is optimized by lowering relative quality of teachers while increasing their numbers. This causes the higher ability college graduates to choose private sector employment which offers higher reward to skilled workers. We show moreover that this loss of the more talented workers by the teaching profession is exacerbated by the teacher wage compression resulting from the teacher union imposed collective bargaining agreements. Further, we argue that a rise in skill premium caused, in particular, by skill biased technological change will deepen the negative trends in the relative quality of education. Indeed, the labor of college graduates will further appreciate relative to the average wage and hence relative to the tax revenue. Countering this trend would therefore require an increase in the share of GDP spent on basic education, assuming that the institutional setup of the school system remains unchanged. Our finding that skill biased technological change can have a negative effect on the quality of education is an interesting case of negative feedback, since SBTC literature points to the rise in the supply of skill due to growing availability of education as its underlying cause. Furthermore, this leads to an issue which appears important for future research on the aggregate long term effects of SBTC: as technical change brings about productivity gains, one needs to factor in its effects on the cost and quality of education and the corresponding policy responses in order to assess the full long-term impact.