IV
MODELS
Part IV addresses issues involved in the modeling of teacher supply, demand, and quality phenomena. The focus is on models developed to project teacher supply and demand variables for use in estimating prospective teacher shortages and surpluses. One purpose is to identify, review, analyze, and compare the most promising of such models that have been developed at the state, regional, and national levels, as well as to review research relevant to modeling teacher supply and demand variables. The range of alternative approaches is described, and the strengths and limitations of the best examples of alternative types are analyzed. Another purpose is to relate the models to information needs of policy makers in dealing with teacher work force issues and to analyze the potential of these models to yield useful information about such issues.
Models for Projecting Teacher Supply, Demand, and Quality: An Assessment of the State of the Art
STEPHEN M. BARRO
INTRODUCTION
Although concerns about future teacher supply and demand seem to be perennial, their content changes to suit the times. The alarms heard not so long ago about an imminent general teacher shortage have receded, to be replaced by increasing attention to the adequacy of prospective supply in science, mathematics, special education, and other particular teaching fields. Discussions of adequacy are now at least as likely to focus on teacher quality as on teacher numbers. The lesson seems to have been absorbed, after much contention over whether there will be "enough" teachers, that quantity per se is not the central problem. Given the willingness to pay and/or sufficient flexibility about standards, we can always hire enough people—and usually enough nominally qualified people—to fill the classrooms. But whether we can find teachers good enough to produce the educational performance gains the nation so urgently needs or to reach the ambitious national education goals that high officials have recently proclaimed are quite different matters. In these respects, the adequacy of the teacher supply is very much in question, and the future supplydemand balance is a major policy concern.
Policy makers' questions about prospects for staffing the schools have stimulated efforts over the years to generate better information on the outlook for teacher supply and demand. Many of these efforts have focused on creating the data bases on which supply and demand analysis necessarily depends—data on the size and makeup of the teaching force, on teacher assignments and career patterns, on persons trained and certificated to teach,
on teacher training institutions and programs, and on the agencies (mainly local school districts) that recruit, employ, and seek to retain teachers. At the same time, other efforts have focused on creating, and then applying, the analytical tools needed to make the data meaningful and to provide policy makers and other users with the information they need—not just the facts, but estimates, inferences, and judgments as to what the facts imply. Prominent among these tools are the teacher supplydemand projection models reviewed in this paper. Ideally, state and federal officials, equipped with such projection models, should be able to monitor and assess developments in the teacher market, estimate trends, and anticipate imbalances or deficiencies (whether quantitative or qualitative) in time to take remedial action. How well these functions can be accomplished with current supplydemand models and how models can be improved to accomplish them better are the principal questions addressed in this assessment.^{1}
Teacher SupplyDemand Models and their Characteristics
A teacher supplydemand projection model consists of a set of mathematical relationships with which future levels of supply, demand, and (in principle) quality can be estimated and (ideally) linked to future economic and educational conditions and policies. Such models have been constructed for particular states by state education agencies and various research organizations: a regional model covering the New England states and New York is under development; and different types of national models have been introduced over the years by the National Center for Education Statistics (NCES). Although these models differ in important respects, all share a common basic structure. A complete teacher supplydemand model consists of three main components or submodels: (1) a submodel for projecting the demand for teachers. (2) a submodel for projecting the supply of continuing or retained teachers (or, equivalently, a model of teacher attrition), and (3) a submodel of the supply of potential entrants into teaching. The latter two components, taken together, should yield projections of the total teacher supply, which one should be able to juxtapose to, and compare with, projections of total demand.^{2} As will be seen, however, many current models offer incomplete treatments, or no treatments at all, of the supply of potential entrants, a situation that seriously limits their usefulness for assessing the overall supplydemand balance.
The types of questions that can be addressed and the types of information that can be generated with a teacher supplydemand projection model depend on several key model characteristics. One is the extent of disaggregation of teachers by teaching field or subject specialty. Some models deal only with teachers in the aggregate (or only with such broad subgroups as
elementary teachers and secondary teachers) and consequently are of no use for assessing the prospective supplydemand balance in particular subject areas. Other models disaggregate in different ways and to different degrees, making it possible to respond to correspondingly more or less detailed policy questions.
A second key characteristic is whether the model is mechanical or behavioral. Mechanical or "demographic" models are capable only of estimating what will happen in the future if established patterns or trends continue. For example, a mechanical model of teacher demand yields estimates of the numbers of teachers that will be required in the future if teacherpupil ratios (or trends therein) remain constant, and a mechanical model of teacher retention predicts how many teachers will remain in the teaching force if the attrition rate for each type of teacher remains unchanged. Behavioral models, in contrast, link the demand and supply estimates to pertinent conditions and policies. Only behavioral models can be used to address whatif questions about the effects of hypothetical changes in circumstances on teacher supply and demand.
A third critical attribute is whether and how the models deal with teacher quality. In fact, nearly all the current projection models focus on numbers of teachers only, avoiding the quality dimension entirely. This makes them at best only peripherally relevant for addressing the qualityrelated teacher supply and demand issues referred to above. Because of the growing urgency of quality concerns, this assessment places special emphasis on the initial tentative steps that have been taken, and the further steps that may be feasible, to take teacher quality into account.
Models also differ in many characteristics of a more technical nature, including explicit and implicit behavioral assumptions, definitions of variables, statistical methods, types of data used, and length of the projection period. The assessment considers how all of the above affect the validity and usefulness of the supply and demand projections.
Purpose and Scope
The general purpose of this assessment is to determine whether the current teacher supplydemand projection models and methods (and some now under development) are well conceived, technically sound, and—most important—capable of satisfying policy makers information needs. More specifically, the assessment addresses the following issues:

How adequate are the present models and methods for estimating future levels of teacher supply, demand, and quality?

How adequate are they for analyzing the effects on teacher supply, demand, and quality of changes in pertinent conditions and policies?

Which current approaches to supplydemand projection modeling appear to be the most promising?

What new or modified models, methods, and data bases might improve the quality of projections and the usefulness of teacher supplydemand analyses?
The assessment covers national models and a selection of models developed by or for individual states. The national models in question are mainly those produced by or for NCES. The statelevel models to be examined were chosen semisystematically from among models cited in the literature, models submitted by states to the National Research Council in response to a general request for teacher supplydemand studies, and models suggested by experts in the field. The coverage of state models is neither comprehensive nor necessarily representative; almost certainly, it is skewed in favor of the more elaborate and sophisticated modeling efforts. The specific state models discussed in this report are those for Connecticut, Indiana, Maryland, Massachusetts, Michigan, Nebraska, New York, North Carolina, Ohio, South Carolina, and Wisconsin. In some instances, however, the available state studies present only certain model components (e.g., only models of teacher attrition for Michigan and North Carolina) rather than complete supplydemand models.
Organization
The main body of this paper is organized around the three model components defined above. Thus, the next sections assess models of the demand for teachers, the supply of retained teachers, and the supply of potential entrants into teaching, respectively. The section on the supply of entrants also covers the approach taken within each model (if any) to analyzing the supplydemand balance. Each section includes a discussion of general issues pertaining to the model component in question; a series of descriptions and evaluations of individual state and national models; and a general assessment of the current state of the art, unresolved problems, and possible avenues of improvement. A final brief section provides an overview of the current state of the art and prospects for improved supplydemand projections in the future.
MODELS OF THE DEMAND FOR TEACHERS
The size of the teaching force in a state or in the nation is determined primarily by how many teachers school systems are able and willing to maintain on their payrolls—that is, by how many teachers the employers demand. Projections of demand are the logical starting points for assessing
the future supplydemand balance. Taken together with estimates of future attrition (discussed in the next section), demand projections indicate how many entrants into the teaching profession will have to be found to replace those who leave, to adjust to changes in enrollment, and to respond to new education policies. Likewise, disaggregated projections of demand by field or subject specialty provide some of the basic information needed to judge whether serious problems are likely to arise, as some have alleged, in finding enough teachers in such areas as science, mathematics, and special and bilingual education. Ideally, demand projection models should also be instruments for assessing the effects of future economic, fiscal, and demographic changes on the size and makeup of the teaching force; examining connections between teacher staffing and teacher compensation; and even exploring tradeoffs between numbers of teachers and teacher quality; but as will be seen, major advances in the state of the modeling art will be needed before any of these broader ambitions can begin to be realized.
Unlike research on teacher attrition, which has progressed rapidly in the last few years, research on the demand for teachers has been minimal. Methods of projecting demand are little changed from what they were when the National Research Council last undertook reviews of teacher supply and demand models (Barro, 1986; Cavin, 1986; Popkin and Atrostic, 1986). The main noticeable advance is that demand estimates now seem more often to be disaggregated, and disaggregated in greater detail, by grade level and subject specialty; however, methods of projecting subjectspecific demand are still rudimentary. Some of the more fundamental improvements called for in earlier reviews—most notably, a shift toward behavioral modeling—have been slow in coming. The importance of making models behavioral is regularly reaffirmed (see, e.g., Gilford and Tenenbaum, 1990), but real behavioral modeling remains, as it was five years ago, a hopedfor future development rather than a reality.
Considerations in Modeling Demand
Before reviewing specific projection models, I discuss briefly some of the major generic issues that arise in modeling the demand for teachers and certain key considerations in assessing present and proposed projection methods.
The Definition of Demand
The number of teachers demanded in a state or in the nation refers, both in standard English and in economics, to the number that school systems want to employ and are prepared to pay for at a given time. The everyday and economic definitions differ, however, in that the former is usually framed in terms of fixed requirements (e.g., predetermined teacherpupil ratios),
whereas the economic definition treats the number of teachers demanded as contingent on such things as how much teachers cost and how much money the employers (school systems) have to spend. The economist, therefore. thinks of demand as a mathematical relationship or schedule—i.e., as a function rather than a number. A demand function relates the number of teachers that employers seek to hire—the quantity demanded—to such factors as the size and composition of enrollment, the level of education funding, and the prices that must be paid for teachers of various types and qualities. According to the everyday definition, a demand projection is an estimate of the number of teachers that will be "required" in the future (e.g., to maintain some stipulated teacherpupil ratio), but, according to the economic definition, it is an estimate of how many teachers school systems will seek to employ, contingent on certain projected values of an array of underlying determinants of demand.
The significance of this definitional difference is that the two concepts lead to different models and modeling strategies. The requirements notion of demand translates into mechanical (sometimes termed demographic) projection models, in which, typically, ratios of teachers to pupils (or trends therein) are assumed to be fixed and projections of numbers of teachers are driven by enrollment forecasts. In contrast, the economic definition points to behavioral models, in which projections derive from multivariate equations linking the number of teachers demanded to various economic, fiscal, and demographic causal factors. Thus far, the mechanical models dominate the field. As a result, the only demands that can now be projected are those that will materialize if no significant changes occur in any of the factors that influence school systems' ability and willingness to employ teachers.
The Relationship Between Demand and Employment
A crucial but often unappreciated consideration bearing on the validity of demand projections is the relationship between the number of teachers demanded and the number actually employed. In principle, both the economic concept and the requirements concept of demand allow for the possibility that the former could exceed the latter—that is, school systems might fail to find the numbers of teachers they are able and willing to pay for.^{3} In practice, however, analysts have almost always assumed (usually implicitly) that the number of teachers demanded is currently the same, and has been the same in the past, as the number actually employed. That is, the prevailing assumption is that the teacher market is characterized by supplydemand balance or excess supply—that there is and has been no excess or unfulfilled demand.^{4}
The validity of demand projections depends strongly on whether the assumption of excess supply is correct. Invariably, the quantities referred to
as demand projections in teacher supplydemand studies are projections of employment rather than demand per se. The implicit assumption is that school systems were able, historically, to find as many teachers as they wanted, and hence a projection of employment and a projection of the number of teachers demanded are one and the same thing. If this were not true—if past levels of employment were determined by numbers of teachers available (supply) rather than by demand—then projections of employment would be supply rather than demand projections. It would become meaningless, then, to speak of the projected supplydemand balance. The distinction between numbers of teachers demanded and numbers actually employed is especially critical when it comes to projecting disaggregated demand by teaching field, for reasons spelled out below.
Disaggregation by Field of Teaching
Recently, concerns about general teacher shortages have receded, and attention has focused instead on the supplydemand balance in particular fields of teaching, especially fields in which difficulty is anticipated in finding enough qualified teachers. To respond to these concerns, many state models offer disaggregated, subjectspecific demand projections. The standard approach to disaggregation is to apply to each subject area exactly the same method as is used to project demand in the aggregate: namely, to multiply projected enrollment in the subject area by an extrapolated, subjectspecific teacherpupil ratio. However, this approach suffers from two serious but often unappreciated conceptual limitations.
First, it presumes that the excesssupply assumption holds for each subject area separately—i.e., that the number of teachers demanded in each field is the same as the number actually employed. This assumption, even if approximately correct for teachers in the aggregate, is less likely to hold for teachers in particular fields. We have been hearing for some time, for example, that there are ''not enough'' teachers in such fields as physics, chemistry, mathematics, and special education. If this is true, it cannot be correct to project future demands for such teachers on the basis of current and past employment. For example, if the number of physics teachers currently demanded were 2 per 1,000 high school students but the number actually employed were only 1.5 per 1,000 because too few qualified applicants were available, then it is logically the former ratio rather than the latter that should be used to project future demand. This is admittedly a difficult prescription to implement because the actual teacherpupil ratio is observable, while unfilled demand is neither observable nor readily inferred. Nevertheless, the issue is unavoidable: it would be logically inconsistent to say that there is excess demand for physics teachers today but then to project the future demand for physics teachers from the number currently employed.
Similarly, subjectspecific demand projections based on historical patterns of course enrollment rest implicitly on the assumption that past course enrollment patterns were wholly demanddetermined rather than jointly determined by both demand and supply. It is quite likely, however, that past levels of enrollment in some subjects were constrained by limited course offerings, which in turn reflected the limited availability of teachers. At the high school level, in particular, the pattern of course taking gives the appearance of being largely demanddetermined, in the sense that it reflects the pupils' own choices, but pupils can choose only among courses that are offered and for which teachers exist. Consider, for example, what would happen if one attempted to project the demand for teachers of Japanese from data on the number of such teachers currently employed. The estimate obtained from such a projection would undoubtedly be very low, but this would reflect the limited present availability of Japanese teachers and the consequent rarity of Japanese as a curricular offering. It seems quite likely that if teachers of Japanese were abundant, many more Japanese courses would be offered, more teachers would be employed, and projected demand would be considerably higher than in the present supplyconstrained situation.
A further obstacle to producing valid disaggregated projections of demand is that neither data bases nor methods have been developed for projecting future rates of course taking by subject. The present models merely reflect the assumption that current or recent distributions of course enrollments by subject area will remain unchanged for the indefinite future—an assumption that is often demonstrably false, given the changes that have been taking place in state curricula and graduation requirements. To my knowledge, no methods of projecting changes in subjectspecific course enrollment rates have yet been demonstrated. It appears, however, that some data bases are now available that would make such projections possible. Certain states (e.g., California and Florida) maintain detailed pupillevel data bases that include information on enrollment by subject and could be used to study changes in course taking over time. National studies of coursetaking behavior based on samples of high school transcripts may also be useful for projecting subjectspecific demands.^{5} This particular unexplored area of demand modeling appears to be ripe for substantial progress.
The Demand for Teacher Quality
Even the most sophisticated projections of numbers of teachers likely to be wanted in the future would not meet the needs of those concerned with issues of teacher quality. The quality dimension of demand has received very little attention (apart from acknowledgments of its omission) in sup
plydemand studies. No demand projection model, to my knowledge, takes any qualityrelated attribute of teachers into account. This situation is unlikely to be rectified soon. Nevertheless, certain aspects of the quality issue are worth addressing, even if for no other reason than to indicate what is missing from, and may be misleading about, projections that do not take teacher quality into account.
States and school districts are clearly not indifferent between teachers of higher and lower quality—however quality might be defined—and there is good reason to believe that districts are willing to pay more to acquire teachers with the qualityrelated attributes they value. For example, some school systems (usually the more affluent ones) generally do not hire inexperienced teachers but instead recruit higherpriced teachers who already have taught elsewhere. Some districts (again, usually the more affluent) appear to offer higher salary schedules than neighboring districts specifically to attract large pools of applicants among whom they can choose. The fact that teacher salary schedules almost universally reward teaching experience and training is evidence in itself that qualifications matter to employers, and that all teachers are not viewed as essentially interchangeable labor. Thus, it is clearly meaningful to speak of a demand for teacher quality, just as one normally speaks of a demand for quantity.
Two related aspects of the demand for teacher quality that merit investigation (and that could conceivably be reflected in a future generation of demand projection models) are (1) the nature of the qualityprice relationship (how much extra are school systems willing to pay for valued attributes of teachers?) and (2) the nature of quantityquality tradeoffs (to what extent would school systems be willing to accept lower teacherpupil ratios to get higherquality but presumably more expensive teachers). The possibility of quantityquality tradeoffs raises serious concerns about projections of numbers of teachers. For instance, a state that has been shifting its emphasis toward higher teacher quality may be employing fewer teachers (but better ones) than would have been projected on the basis of earlier data. With a demand model that takes no account of quality or price, such a development would either be missed or misconstrued. Moreover, even a behavioral demand model would yield misleading results if it lacked a quality dimension; for example, a tradeoff of quantity for quality could be misinterpreted as a reduction in the number of teachers demanded in response to rising cost. Interstate comparisons of teacher demand also remain problematic without methods of taking quality differentials into account.
Ideally, one can envision a model that projects both the number of teachers and the quality of teachers demanded in a state, but the obstacles to creating such an analytical tool are formidable. Quality measurement is only one major problem. It is actually less of a problem in connection with demand projections than in other contexts because what counts in a demand
analysis is only whether employers are willing to pay for a given teacher characteristic, not whether that characteristic is truly educationally valuable. For example, if districts are willing to pay extra for teachers with master's degrees, then having a master's is an appropriate element of quality to include in a demand model regardless of whether master's degrees translate into improved learning for pupils. But even assuming that teacher quality can be measured, there are still difficulties in characterizing, much less projecting, the quality of a teaching force. One is that states employ mixes of teachers with different characteristics and quality, not teachers of uniform quality, which makes it difficult to characterize any particular state's quality or to compare quality levels among states. Another is that each state's teaching force was accumulated over a period of many years, which means that the average quality of today's teaching force reflects both the demand for and the supply of teacher quality in many past periods. The value that the state currently places on quality is reflected only in hiring new teachers. Moreover, the quality of retained teachers reflects demandside influences but also such supplyside phenomena as the relative rates at which lowerquality and higherquality teachers leave the profession and the propensities of teachers to improve their own quality through education and onthejob learning. Thus, there is much to sort out before even beginning to model or project the demand for teacher quality.
National and State Demand Projection Models
The Standard Demand Projection Model
With few exceptions, projections of the demand for teachers are made according to a simple, mechanical, standard model. Each version of this standard model projects the future demand for teachers (either in the aggregate or in a particular category) as:
Projected number of teachers demanded in year t 
= 
Projected enrollment in year t 
× 
Projected teacherpupil ratio in year t 
where the projected number of teachers demanded refers either to all teachers or to teachers at a particular level (elementary or secondary) and/or in a particular subject area; the projected teacherpupil ratio refers to the same level and/or subject category; and projected enrollment is either aggregate enrollment, enrollment at the specified level, or course enrollment in the specified subject area, as the case may be.
The enrollment projections that enter into these models are generally
produced separately and treated as inputs into the teacher demand calculations.^{6} Many state education agencies routinely maintain and operate enrollment projection models for reasons unrelated to concerns about teacher supply and demand (e.g., to develop and justify budget requests), and NCES provides regularly updated national enrollment projections and, of late, enrollment projections by state.^{7} The accuracy of demand projections naturally depends on the accuracy of the underlying projections of enrollment. Whether the enrollment projection methods deal adequately with migration, dropping out, and private school enrollment are particularly sensitive concerns. To cover these issues adequately, however, would require a specialized review of enrollment projection methodology—a task beyond the scope of this paper. From here on, I simply take enrollment projections as given, leaving it to others to assess their quality.
Projections of future teacherpupil ratios are generated, in the standard model, by one or a combination of the following methods:

Method 1: Assume that teacherpupil ratios in future years will be the same as in the most recent year for which data are available.

Method 2: Assume that teacherpupil ratios in the future will be equal to the averages of the corresponding ratios in several recent years.

Method 3: Estimate future teacherpupil ratios by extrapolating the trends of recent years.

Method 4: Set the projected teacherpupil ratios to reflect adopted or anticipated future staffing norms or staffing policies.
There is little scope for creativity within the framework of the standard model. The principal choices to be made by each model developer concern (1) the number and type of teacher categories for which separate demand projections will be developed, (2) which of the four methods listed above will be used to project teacherpupil ratios, and (3) the number of future years for which projections will be made and the number of past years on which the projections will be based.
Variations on a Theme
The following are brief descriptions of how several states have implemented the standard model and how NCES used it to make national projections prior to introducing a new methodology in 1988. The descriptions are followed by remarks on the limitations of this general approach to projecting teacher demand.
Nebraska A Nebraska model (Ostrander et al., 1988) illustrates the standard model in its simplest form. It projects demands for elementary and secondary teachers (grades K6 and 712, respectively) for the years 1988–
89 to 1993–94 by multiplying projected enrollment for each level in each year by the corresponding teacherpupil ratio in the most recent year, 1987–88, for which data were available. In essence, this model simply assumes that the number of teachers wanted at each level will increase each year in proportion to the annual increase in the number of pupils to be served.
Wisconsin A Wisconsin analysis (Lauritzen and Friedman, 1991) projects demand for the threeyear period 1991–92 to 1993–94 on the basis of enrollment projections prepared by the Wisconsin Department of Public Instruction. It then extends the projections two additional years into the future (i.e., past the period for which enrollment projections are available) by making broad assumptions about rates at which enrollments will continue to grow. The model uses fixed teacherpupil ratios (actual ratios in the 1990–91 base year) to project demands for elementary, secondary, and special education teachers. Thus, like the Nebraska model, it reflects the assumption of simple proportionality between the number of teachers at each level and the number of pupils enrolled. (Interestingly, the same Wisconsin study also offers more detailed projections of future numbers of newly hired teachers, covering 16 subjectarea categories of regular education teachers and 8 categories of special education teachers, but these are not based on projections of numbers of teachers demanded but rather on direct extrapolations of numbers of new hires in previous years.)
New York A model developed by the New York State Education Department (1990) projects demand for teachers in 15 subject specialties grouped into elementary (K6), secondary (712), and combined elementarysecondary (K12) levels. The projected demand for each type of teacher for each year from 1989–90 to 1993–94 is determined by multiplying projected enrollment for the appropriate level by the projected teacherpupil ratio for the subject area in question. Separate projections are prepared for New York City and for the remainder of the state, and the two sets of figures are summed to produce estimates for the state as a whole. According to the New York report, most teacherpupil ratios were assumed to remain constant at their baseyear (1988–89) values throughout the projection period, but "for a few highgrowth subjects, the ratios were allowed to [increase] slightly for a year, and then held constant." Note that this method allows for no changes in the enrollment mix by subject area—that is, enrollments in secondary mathematics, secondary English, secondary occupational education, etc., are all assumed to change over the years in the same proportion as total secondary enrollment.
South Carolina The South Carolina Department of Education (1990) projects the 1995 demand for teachers in 36 different subject specialties on the basis of teacherpupil ratios prevailing in 1990. (For some reason, projections for the intermediate years are not provided.) The model is driven by enrollment projections for each of six levels: child development,
kindergarten, other early childhood, elementary, junior high, and high school. Specifically, the projected 1995 demand for teachers in each subject category is determined for each level by multiplying the actual number of teachers in the category in 1990 by the ratio of projected 1995 enrollment to actual 1990 enrollment at the level in question. The results are then summed over levels to determine the total demand for teachers of each subject. This procedure is mathematically equivalent to assuming that all teacherpupil ratios will remain fixed and that course enrollments in all subjects offered at a given level will increase in proportion to the increase in that level's enrollment. In other words, this model, like the New York model, imposes the rigid assumption that no change will occur during the projection period in the mix of subjects taken by the pupils at each level.
Ohio An Ohio model (Ohio Department of Education, 1988) differs from the models outlined above in that it projects the teacherpupil ratio in each subject area according to a trend extrapolation. The model uses data for school years 1982–83 to 1986–87 to project demands for teachers in 29 subject areas over the fiveyear period 1987–88 to 1991–92. The teacherpupil ratio for a particular subject area is projected, in most instances, by fitting a leastsquares regression line to data for the fiveyear base period. In a few cases, timetrend extrapolations are deemed inappropriate, and teacherpupil ratios are set at baseperiod averages or other stipulated values. These projected teacherpupil ratios are then applied to projected enrollments for the appropriate grade levels. As in the models described previously, no allowance is made for the possibility that the mix of courses taken by pupils at a given grade level will change during the projection period.
Massachusetts A teacher supply and demand analysis and projection model for Massachusetts has been produced by a group at the Massachusetts Institute for Social and Economic Research (MISER), based at the University of Massachusetts (Coelen and Wilson, 1987). Unlike the other modeling efforts described above, this one seeks to provide a simulation tool rather than a single set of numerical projections. In pursuit of this goal but faced with severe data constraints, Coelen and Wilson have created a more complex conceptual structure than they have been able to implement empirically. On the demand side, the model features a full matrix of course enrollment rates by grade level in each of 20 subject areas, which would allow, in principle, for varied and flexible assumptions about trends in subjectarea enrollments. But the available coursetaking data for Massachusetts, unfortunately, are so skimpy that Coelen and Wilson could do no better than to assume the same fixedproportion course enrollment patterns as underlie more modest models.^{8} Also on the demand side, the part of the model that deals with teacherpupil ratios reflects no particular theory about how those ratios will be determined in the future but rather is intended to
answer whatif questions about alternative possibilities. Consequently, the model's basecase assumption becomes, by default, essentially the same as underlies many other states' projections—namely, that the teacherpupil ratio in each subject area will stay constant in the future at its current or recent value.^{9} In these respects, the Massachusetts demand model, though more elaborate than the others, remains in the standard model category.
The analysts at MISER have been engaged for some time in a much more ambitious project intended to produce teacher supply and demand information, including projections, for a multistate region encompassing the New England states and New York. This project, known as the Northeast Educator Supply and Demand study (NEDSAD), has amassed large, multiyear data bases for each participating state, and efforts to develop the supply and demand projection models are now under way. Coelen and Wilson have said that they intend these models to be "econometric," or behavioral (as contrasted with the "demographic," or mechanical, models presented in the Massachusetts study), but it appears, at this point, that this applies to the supply side of the models and not to the projections of demand. My understanding, based on preliminary model specifications (Coelen and Wilson, 1991a) and a discussion with Coelen, is that demand projections in the new model, as in the earlier Massachusetts model, will continue to be driven exclusively by projected enrollment and coursetaking patterns. Moreover, the present MISER plan apparently is to develop demand models using only data aggregated to the state level, whereas a districtlevel analysis would be needed to capture relationships between the demand for teachers and multiple economic, fiscal, and demographic factors. (Note: remarks on the supply components of the NEDSAD model appear in later sections.)
Connecticut Connecticut's approach to projecting demand (Connecticut Board of Education, 1988) conforms in most respects to the standard model but also includes two useful extensions: (1) a mechanism to allow for nonproportional responses of the demand for teachers to changes in enrollment and (2) an explicit provision for adjusting demand projections for changes in staffing policies. The model projects demands for three gradelevel groupings (elementary, middle/junior high, and high school) and eight staff categories: six subjectarea categories of classroom teachers plus a support staff category and an administrator category.^{10} It provides projections for the 14year period from 1988–89 to 2000–2001, based on data for the years 1986–87 and 1987–88.
The Connecticut model calculates the projected yeartoyear change in the demand for staff in each category and at each level as the mathematical product of (a) the projected enrollment change for the level in question, (b) the applicable teacherpupil ratio, and (c) a response rate, which represents the rate at which Connecticut school systems tend to increase their staffs in response to each unit increase in enrollment. The set of response rates is a
distinctive feature of the Connecticut model. These rates vary by staff category and level, but generally have values less than 1.0, signifying that, although staff hiring responds to changes in enrollment, it responds less than in proportion. Moreover, the response rates take on higher values for increases in enrollment than for decreases, reflecting evidence that district hiring responds more strongly to enrollment increases than to enrollment declines. The values of all the response rates were determined by comparing rates of staff change with rates of enrollment change between the years 1986–87 and 1987–88.
The model also allows explicitly, as noted above, for changes in staffing due to changes in education policies, but these are determined on an ad hoc basis. For instance, the Connecticut analysis allows for phasing in of anticipated staffing increases attributable to expected increases in the number of Connecticut districts offering fullday kindergarten programs.
The computations of projected changes in demand are repeated year by year for each year of the projection period. Because of both the responserate factors and the allowances for effects of policy changes, the teacherpupil ratios used in the projections change from one year to the next. Were it not for the enrollment and policy changes, however, the ratios would remain constant; that is, there is no assumption of any underlying trend.
The responserate feature of the Connecticut model reflects the findings of a research paper (Murnane et al., 1985) on district responses to upward and downward changes in enrollment. It should be noted, however, that the projection model does not take into account one important finding of that research—namely, that there is a significant difference between shortrun and longrun responses of staffing levels to enrollment changes, especially in the case of falling enrollment. The distinction between immediate responses and longterm adjustments could be important, especially following a period of enrollment decline, and projections might be more accurate if it were taken into account.
The NCES Pre1988 National Model NCES has produced projections of the demand for teachers for many years, but its projection method changed drastically in 1988. The current model, described later, uses a regression equation to relate projected demand to several causal factors (Gerald and Hussar, 1990). The pre1988 model is a version of the standard model outlined above—that is, it projects demand as the mathematical product of projected enrollments and projected teacherpupil ratios. Specifically, NCES, as of 1985, produced socalled low, intermediate, and high projections of numbers of elementary and secondary teachers by multiplying the projected enrollment at each level by teacherpupil ratios determined as follows:

For the low projection, assuming that the teacherpupil ratio would remain constant at its value in the most recent year for which data were available;

For the high projection, extrapolating the trend in the teacherpupil ratio according to an exponential smoothing technique, wherein the historical data are weighted according to an exponential function so that more recent observations receive greater weight; and

For the intermediate projection, setting the teacherpupil ratio at the average of the high and low projections (Gerald, 1985).
Thus, the former NCES approach actually blended two variants of the standard model: one reflecting the assumption of a constant teacherpupil ratio; the other, the assumption of a continuing trend.
NCES Projections of Numbers of Teachers by State Although NCES has shifted to a regressionbased model for its national demand projections. it has adhered to the standard model in producing a set of teacher demand projections by state (Gerald et al., 1989b). The statelevel projection model is essentially the same as the pre1988 NCES national projection model: the number of public classroom teachers in a state (elementary and secondary combined) is projected for fall 1988 to fall 1993 by multiplying the state's projected K12 enrollment by a projected teacherpupil ratio. The latter is obtained by extrapolating the trend in the state's teacherpupil ratio from 1970 to 1987 according to the aforementioned exponential smoothing technique.
General Assessment of Projections Based on the Standard Model
The limitations of the demand projections described above are for the most part selfevident and have been discussed in earlier reviews, so I summarize them only briefly here.
First, although the mechanical projection models reflect no explicit assumptions about future conditions affecting teacher demand, they implicitly reflect the assumption that all major influences on demand will either remain constant or continue to change at the rates at which they have been changing in the past. If anything substantially different were to happen—for instance, if state or local fiscal conditions were to change or if the relative salaries of teachers were to rise or fall sharply—the projections obtained from such models would probably be incorrect.
Second, although the projection models can produce teacher demand estimates corresponding to different stipulated teacherpupil ratios and different patterns of enrollment by subject, the models themselves have no capability to predict how these ratios or patterns would be affected by policy changes or other external developments. Consequently, they cannot be used (except in conjunction with ad hoc outside analyses) to respond to whatif questions about the effects of such changes on teacher demand.
Third, although some models break down projections of teacher de
mand by subject area, they do so only by assuming that the mix of course enrollments by subject area will remain the same in the future as it is today. They make no allowances for disparate trends in course taking in different fields or for specific developments, such as changes in curricula and graduation requirements, that are causing coursetaking patterns to change.
Fourth, an implicit assumption underlying the projections of teacher demand by subject area is that the market for each type of teacher is and has been in a condition of excess supply. Although this assumption is realistic for teachers in the aggregate, it may not be valid for certain specific fields, such as chemistry, physics, and special education. If it is not, projected demands for teachers in those fields would be understated.
Fifth, an assumption built into the standard model is that the demand for each type of teacher responds proportionately to any increase or decrease in the corresponding category of enrollment. Evidence has shown, however, that the actual response is likely to be less than proportional, especially in the short run. Of the models examined, only that of Connecticut allows for these nonproportional responses.
Finally, even taking the models on their own terms—that is, accepting their mechanical nature and assuming that the builtin assumptions are valid—there are still certain possibilities for improvement. The models that base projections only on data for the most recent year might benefit from using multiyear data to quantify trends in the teacherpupil ratio. Similarly, models that assume permanently fixed patterns of course taking by subject might be upgraded by projecting separate trends in enrollment by subject area.
An NCES Venture into Econometric Modeling
In 1988, NCES switched from the standard projection model described earlier to a new econometric approach. In the three most recent editions of its Projections of Education Statistics, the agency has used regression equations fitted to national time series data to predict future numbers of public elementary and secondary teachers (Gerald et al., 1988, 1989a; Gerald and Hussar, 1990). This regression model, as it now stands, is a very simple one with some serious conceptual and technical flaws; yet it represents an important first step, and one to be encouraged, away from the traditional mechanical projections and in the direction of behavioral models.
Model Description
The NCES regression equations for elementary and secondary teachers both relate the number of teachers demanded to enrollment, per capita income, and state education aid per pupil. The equation for elementary teachers, as presented in Gerald and Hussar (1990) is:
ELTCH = b_{0} + b_{1}PCI_{2} + b_{2}SGRANT + b_{3}ELENR
and the equation used to project the number of secondary teachers is:
SCTCH = b_{0} + b_{1}PCI_{2} + b_{2}SGRANT_{3} + b_{3}SCENR
where ELTCH and SCTCH are the numbers of public elementary and secondary teachers, respectively; PCI2 is disposable income per capita (adjusted for inflation), lagged 2 years; SGRANT is local education revenue receipts from state sources per capita (adjusted for inflation): SGRANT3 is the same variable lagged 3 years; and ELENR and SCENR are public elementary and secondary enrollment, respectively. NCES used ordinary leastsquares regression to fit these equations to 30 sets of annual observations. The particular functional form and the different data lags were arrived at, apparently, by trying various specifications and selecting the one with the best statistical properties. The regression equations presented in the immediately preceding edition of Projections (Gerald et al., 1989a) resemble the foregoing equations very closely, differing only in the time lags in the PCI and SGRANT variables.^{11}
To use such equations for projecting numbers of teachers, one must, of course, have projections of all the independent variables. Projections of the two enrollment variables, ELENR and SCENR, are obtained from the NCES enrollment projection models described in the same annual Projections volumes. Projections of disposable per capita income (PCI) are derived from a macroeconomic model created by Data Resources, Inc. (DRI). The projections of the state grant variable (SGRANT) are produced simply by assuming that state grants per pupil will continue to increase at the same percentage rate as during the three most recent years, namely, by 3.1 percent per year. Using these inputs, NCES generates estimates of numbers of public elementary and secondary teachers for each year of the projection period (1989–90 to 2000–2001, in the case of the most recent NCES report). In addition, NCES projects private school teacher demand simply by assuming that the numbers of private elementary and secondary teachers will increase at the same rates as estimated for their public school counterparts.
Assessment
At this point, the new NCES regression method is less valuable for what it is than for what it may bring forth. In the hope that further development of the econometric approach will occur. I offer the following critique and suggestions for improvement.
From a theoretical perspective, the regression equations presented by NCES represent only part of a model of the demand for teachers rather than a complete theorybased specification. For example, although the models
do allow for the effect of per capita income on demand, they leave out the effects of the pupilpopulation ratio and the price (relative salary) of teachers, even though the relevance of both is well established in the literature. A full theoretical model could be specified in either of two ways, depending on which economic and fiscal variables are to be taken as given for the purpose of projecting demand. One approach is to treat the level of education expenditure per pupil as the major fiscal determinant of the demand for teachers and, accordingly, to express demand as a function mainly of expenditure level and price. Perpupil expenditure itself would have to be projected first, presumably as a function of per capita income, the pupilpopulation ratio, some indicator of the relative price of educational services, and perhaps other factors. The second approach is to merge the two stages of the aforesaid model into one, as Gerald and Hussar (1990) apparently intended to do with their regression equations. In the merged version, the demand for teachers would be expressed as a function of per capita income (or some other fiscal capacity indicator), the pupilpopulation ratio, the relative price of teachers, and possibly some additional factors.
An important conceptual shortening of the present GeraldHussar formulation is that it treats state aid to local school systems (SGRANT) as an exogenous variable instead of as a factor that itself needs to be explained. Although it is certainly true that state aid is a major determinant of spending by local school districts, the objective in this instance is to explain the overall level of spending, and hence teacher demand, in the nation. Treating the largest component of school spending, state aid, as exogenous and projecting it mechanically merely avoids the issue. If state aid does need to appear in the teacher demand function (or in the equation for estimating per pupil spending), then it too should be projected as a function of underlying economic and demographic variables.
Turning to modeling strategy, I question for several reasons whether fitting the teacher demand equations to a 30year set of national timeseries data is reasonable. First, 30 data points would be too few to fit fully specified teacher demand or expenditure models. Second, the assumption of a stable relationship between the education and fiscal variables extending over 30 year is difficult to accept, especially considering that good measures are not available of changes in the relative price of education over that extended period. Third, it seems to me that there is a better alternative: the model can be fitted to pooled timeseries, crosssection data for states. There is no reason why the demand equations, even though they are to be used for making national projections, need to be estimated with nationalaggregate data. Using statelevel data (for, say, the last 5 to 10 years) would permit development of a more detailed model, while avoiding the strained assumption of stable public tastes for education over three decades. Although a statelevel analysis would, admittedly, be a much larger en
deavor than the present national timeseries analysis, it would also yield richer, more useful, and probably more valid results.
On the technical level. I believe that it is misleading to express the dependent variables of the equations in absolute terms (i.e., actual number of teachers) and to use enrollment as one of the explanatory variables. Doing so creates the illusion of high explanatory power (high R2) simply because of the correlation between number of teachers and number of pupils; at the same time, it obscures the roles of the other explanatory variables. I would treat the teacherpupil ratio as the dependent variable and express all the righthand side variables in perpupil or per capita terms. I also believe that the price deflators in the model need to be reconsidered. Assuming that a relative price variable appears in the model, as it should, the appropriate specification would be to deflate the education variables (spending and state aid per pupil) by a measure of education cost and to deflate per capita income by a general price index.
In sum, a great deal can be done to develop the strategy of econometric projection modeling opened up in the 1988 to 1990 editions of the NCES projections. If similar models could be developed for a few individual states, so much the better. This is the general approach that needs to be followed if we are ever to have behavioral models for assessing teacher supply and demand.
MODELS OF THE SUPPLY OF RETAINED TEACHERS
The preponderant share of a state's teaching force in any school year—typically, 92 to 96 percent of the total—consists of teachers retained from the previous year. The ability to estimate the future numbers of these continuing teachers, and hence the number of new teachers to be hired, depends on the accuracy with which retention or attrition rates can be measured and predicted. Even a small percentage error in projecting retention translates into a large error in the estimated demand for new hires. For example, a change in a state's teacher retention rate from 96 to 95 percent corresponds to an increase in attrition from 4 percent to 5 percent and hence a 25 percent increase in the number of entrants needed to maintain the size of the state's teaching staff. There is considerable practical motivation, therefore, for efforts to improve and refine teacher attrition/retention models.
When I and others reviewed teacher supply and demand projection methods for the National Research Council in 1986 (Barro, 1986; Cavin, 1986; Popkin and Atrostic, 1986), we found attrition models generally to be crude. Many states projected the supply of retained teachers simply by applying undifferentiated statewideaverage attrition rates to all categories of teachers in all future years. NCES did the same thing in making its national projections. The states of New York and Connecticut, however, had made the major leap
forward of basing projections on disaggregated agespecific and subjectspecific attrition rates. That, in 1986, was the leadingedge methodology. There were at that time no behavioral models of attrition, no models of change in attrition rates over time, and, of course, no models dealing with the quality as well as the numbers of retained teachers.
Has much has the state of the art advanced? The answer differs sharply depending on whether one considers practical supplydemand projection models only or also takes into account the recent academic research on teacher attrition. Judging only by the explicit projection models, one would have to conclude that little progress has been made. Some states still use a single, undifferentiated, average attrition rate to estimate numbers of continuing teachers. NCES, in its national model, has advanced only from using a single average rate to using separate average rates for elementary and secondary teachers. New York and Connecticut, now joined by Massachusetts and possibly others, disaggregate attrition rates by age and subject specialty as they did before. Otherwise, no major new features are apparent, at least not in the lessthanrepresentative sample of state models I have examined.
In sharp contrast, progress in research on teacher attrition has been dramatic. Several scholars, most notably Richard Murnane of Harvard University and his colleagues and David Grissmer and his associates at the RAND Corporation, have developed far more sophisticated models of teacher attrition and other aspects of teachers' career paths than existed just a few years ago. Using largescale, multiyear state data bases and the statistical tools of survival or hazard modeling, they have been able to address some of the major hitherto unresolved issues of teacher attrition. Thus, we now have multivariate behavioral analyses of attrition, some initial analyses of changes in attrition patterns over time, and even, in Murnane's work, an analysis of the relationship of attrition to an indicator of teacher quality. Although these analyses were not conducted specifically to develop projection models, some of the key research findings and methods seem eminently suitable for that purpose. Accordingly, I deal in this review with selected research on teacher attrition as well as with explicit state and national projection models. First, however, I comment on some general considerations pertaining to attrition rates and their uses in projecting the supply of retained teachers.
Considerations in Modeling Attrition
The following issues must all be dealt with, explicitly or implicitly, in constructing a projection model, and all should be kept in mind in assessing the specific attrition models described later in this chapter.
Defining and Measuring Attrition Rates
An attrition rate is the fraction or percentage of teachers employed in one period (either in some specified teacher category or in all categories combined) who are not employed as teachers in a subsequent period. A retention rate is the complement of the corresponding attrition rate—that is, retention rate = 1attrition rate or, in percentage terms, retention percentage = 100attrition percentage. But although the basic definition is clear, at least two of its aspects require attention: (1) the unit of analysis or level of aggregation and (2) the treatment of individuals who leave teaching for a time but not permanently. (A third aspect, the degree of disaggregation of teachers by level and subject specialty, is discussed separately below.)
The unitofanalysis issue is significant because teachers often transfer among jurisdictions. The average attrition rate experienced by the local school districts in a state is greater than the statewide attrition rate because some teachers leaving particular districts are hired by other districts in the same state and are not lost to the state as a whole.^{12} Similarly, the average state attrition rate is greater than the national rate because some teachers leave one state to teach in another. The fact that some local and state attrition reflects interjurisdictional transfers has implications for both measurement and modeling. The implication for measurement is that one cannot determine a state's attrition rate accurately simply by adding up the numbers of teachers that the state's individual districts or schools have lost. Districts usually are not well equipped to distinguish between teachers transferring and teachers leaving the profession.^{13} Statewide data on individual teachers are generally required to measure a state's attrition rate properly. For the same reason, one cannot measure national attrition correctly by adding up attrition reported by states: a national data base is required, such as that provided by the NCES Schools and Staffing Survey (SASS), applications of which are discussed later.
Transfers between public and private schools also complicate the task of quantifying attrition. Because some teachers leave private schools to take jobs in public school districts (and vice versa), the total number of teachers leaving a state's teaching force in a given year is less than the sum of leavers from the public and private sectors. An analysis of total statewide attrition (public and private combined) would have to be based, therefore, on a data file that includes both public school and private school teachers. In practice, however, most teacher supplydemand models pertain only to the public schools, which means that attrition from private schools is not taken into account and transfers from private to public schools are treated as a component of the supply of ''new'' public school teachers.
The implications of the transfer phenomenon for modeling are that (1) it is important to distinguish, in modeling attrition, between teachers who
transfer and teachers who leave the profession and (2) it may be equally important, in modeling the supply of entrants, to distinguish between transferring teachers and other new hires. Each teacher who transfers from one state (or district) to another counts not only in the former state's (or district's) attrition but also in the latter's new teacher supply. Projecting this component of supply is important wherever there is significant importation of teachers into a state. It is particularly important in constructing a multistate model of teacher supply and demand, such as the NEDSAD model now being developed by the MISER group in Massachusetts. But developing importation projections, it turns out, is one of the more difficult and least explored tasks in modeling teacher supply.
The distinction between temporary and permanent (or between shortterm and longterm) attrition is of considerable importance both to policy makers and to those attempting to project teacher supply. On the policy side, a state's decisions about such things as how many new teachers to train and what measures to take to attract teachers from other sources could be affected by information about the rate at which departing teachers are likely to return. Several recent studies have shown that substantial percentages of each year's entering teachers (i.e., teachers employed in a given year who were not teaching in the previous year) are, in fact, former teachers reentering the profession (Grissmer and Kirby, 1987, 1991; Coelen and Wilson, 1987; Connecticut Board of Education, 1988).
The question of how reentering teachers should be treated in modeling attrition is not yet resolved. Grissmer and Kirby (1987, 1991) distinguish between "annual" and "permanent" attrition. The annual attrition rate is the fraction of teachers employed at the beginning of one year who are not employed at the beginning of the next, including teachers who eventually return; the permanent attrition rate is the (significantly lower) fraction of teachers who leave and do not return, at least during the period for which data are available. Grissmer and Kirby (1991) present separate analyses of attrition patterns and determinants of attrition corresponding to the two definitions. In contrast, most other analysts deal with the reentry problem by analyzing only what Grissmer and Kirby call annual attrition but then modeling the reentry rate as a separate phenomenon (e.g., Murnane et al., 1988, 1989; Connecticut Board of Education, 1988; Coelen and Wilson, 1987). Each approach, thus far, seems to have significant limitations. The notion of permanent attrition is flawed in that (1) it lumps together in the nonpermanent category teachers who leave for only 1 year with teachers who leave for as much as 5 or 10 years and (2) it requires, as a practical matter, either an arbitrary cutoff (e.g., teachers who do not return within 5 years) or retrospective longitudinal data that cannot be collected in time to be useful for projections. The alternative of treating all leavers alike and then modeling reentry separately has the potential weakness of lumping
together persons who may be leaving teaching for very different reasons—e.g., teachers going on oneyear leaves with the intention of returning and teachers leaving permanently for higherpaying occupations. Thus, it remains unclear at this time how the dual phenomena of shortterm attrition and reentry might best be quantified and modeled.
Voluntary versus Involuntary Attrition
In most supplydemand modeling efforts, the supply of retained teachers is taken as synonymous with the number who actually remain in their jobs from one year to the next, the implicit assumption being that all prioryear teachers who want to continue teaching are retained by their employers. The possibility that the supply of wouldbe retainees might be greater than the number actually employed is rarely, if ever, allowed for in projections, even though the number supplied and the number employed are two different concepts. The former might exceed the latter for two reasons. First, sharp cuts in demand (due, e.g., to enrollment declines and/or budget cuts) might lead local LEAs to lay off teachers who want to continue working. (Cuts must be quite sharp to have this effect, considering that some teachers leave on their own each year.) Second, some prioryear teachers might be discharged for other reasons (e.g., incompetence) and replaced with new hires, even though they want to continue teaching. The latter is very rare (although not unheard of) and probably would have little effect on state or national projections. Statewide reductionsinforce have definitely occurred in response to both demographic and fiscal developments, however, and may recur in the future. For instance, Coelen and Wilson (1987) cite sharp staff cutbacks following the enactment of Proposition 21/2 (a statewide tax limitation provision) in Massachusetts in 1981. Therefore, although it is reasonable to assume in most instances that the number of prioryear teachers actually employed in a given year is roughly the same in the aggregate as the number willing to supply their services, this assumption is sometimes violated, and projections based on it may confound supplyside and demandside effects on teacher retention.
In principle, a clear distinction should be made between voluntary and involuntary attrition in modeling the supply of retained teachers. Voluntary attrition is a supplyside phenomenon that reflects the decisions of individual teachers. It makes sense, therefore, to project voluntary attrition as a function of individual teacher characteristics and variables that may affect teachers' decisions (salaries, working conditions, etc.), as is done in some of the attrition models discussed later. Involuntary attrition is a demandside phenomenon that reflects decisions by employers. Mixing together teachers who decide to leave on their own with teachers who are laid off or
fired can lead to incorrect inferences about patterns of, and influences on, attrition and hence to incorrect projections of future attrition rates.
Although most model developers are aware of the problem, none has yet invented a method of dealing with it. The prevailing approach seems to be to rely as little as possible on data for periods in which involuntary terminations are believed to have been numerous. Coelen and Wilson (1987), for instance, do not allow data for the period when Proposition 21/2 took effect to influence their attrition projections, and Murnane and Olsen (1990), in their analysis of North Carolina, place more emphasis on years of growing enrollment and demand for teachers (1975–79) than on years of contracting demand (1980–84). Thus far, the issue remains unresolved of whether it is feasible to distinguish analytically between demandside and supplyside influences on attrition.
Disaggregation by Subject Specialty
Projections of the supply of retained teachers must be disaggregated into the same categories as projections of demand to produce the subjectspecific assessments of the future supplydemand balance that policy makers want, but disaggregation creates additional problems for attrition modeling and adds to some of the problems already discussed. Among the principal complications are the following.
First, measuring attrition rates and numbers of teachers by field is not a straightforward matter because many teachers have certificates in multiple fields. According to Coelen and Wilson (1991b), the average Massachusetts teacher holds 3.3 certifications.^{14} To the extent that teachers are multiply certificated or multiply qualified, the sum of teacher supply by field will exceed teacher supply in the aggregate. For instance, an LEA might have 100 mathematics and science teachers, of whom 40 are certificated to teach mathematics only, 35 to teach science only, and 25 to teach both mathematics and science. It could be misleading, then, to project the retained supply of, say, science teachers by applying a retention rate to the number of teachers actually teaching science in a particular year. Such a projection would understate the supply (or potential supply) of science teachers because some individuals teaching mathematics are in the supply of science teachers as well. Although most developers of supply and demand models recognize this problem, the usual response is to circumvent it by such expedients as classifying teachers only according to the main subject that they actually teach in the base year of the attrition calculations. The actual supply of teachers in any given field is likely to be considerably larger than such calculations indicate. Also, the overall teacher supply is likely to be considerably more flexible than is implied by a method that rigidly assigns each teacher to one and only one field.
Second, disaggregation by subject creates a major new category of transferring teachers: those who change subject areas between one school year and the next. An individual who transfers from teaching mathematics to teaching science within the same LEA (or state) presumably counts as a departing mathematics teacher for the purpose of measuring the attrition rate in mathematics and, at the same time, as an entering science teacher. In the aggregate, however, such a teacher is a retainee. Some special treatment of intersubject transfers is required, not only to reconcile the aggregate attrition rate with the subjectspecific attrition rates but also, more importantly. because different causal factors are undoubtedly associated with shifts among subject fields than with exits from the teaching force. Very little is said in the reports I have seen about how such transfers are handled; the topic appears to qualify, at present, as one of the loose ends in projection methodology.
Third, disaggregation by subject has the effects of (1) amplifying the importance of distinguishing between voluntary and involuntary attrition and (2) undercutting the key assumption of attrition modeling that the number of retained teachers in each category is less than the number of teachers demanded. Although the latter assumption may be only infrequently violated in the aggregate, it is likely to be violated much more often in individual subject fields. Even when enrollments and education budgets are stable, changes in curricula and graduation requirements or in the makeup of the pupil population may cause the demands for course enrollments in particular fields to decline while demands for other fields are rising. Thus. there may be too little demand for teaching in a particular field to justify retaining all the prioryear teachers in that field who want to continue working. As academic requirements for high school graduation are strengthened, for example, the demand for vocational education courses may decline, and hence vocational teachers may have to be laid off or transferred to other fields even though total high school enrollment is rising. Under such conditions, conventional methods of quantifying attrition would confound voluntary with involuntary attrition, and supply projections based on the resulting attrition rates would be misleading. Again, the major unresolved issue—this time in the context of fieldspecific rather than aggregate attrition—is whether it is possible to distinguish analytically between demandside and supplyside determinants of the attrition rate.
Attrition Rates in Relation to Attributes of Teachers
A major characteristic that distinguishes more refined projections of the supply of retained teachers from crude ones is disaggregation of attrition rates by age, subject specialty, and perhaps other attributes of teachers. Several aspects of the pattern of these disaggregated rates are now firmly
established in the literature (the specific studies on which these findings are based are described later in this section.
First, the relationship between attrition rate and age is Ushaped. Young (or new) teachers and older teachers (age 55 and over) leave at high rates; those in between (especially in the 35 to 55 age range) leave at much lower rates. The key implication for projection modeling is that the overall rate of teacher attrition in a state, or in the nation, is likely to change over time simply because of the changing age composition of the teaching force, even if there is no change in the propensity to leave of the teachers in each age bracket. Models that neglect this agecomposition effect on attrition are unlikely to yield accurate estimates of numbers of future retainees.
Second, there are substantial differences in attrition rates among subject specialties. Several studies have shown that secondary teachers tend to leave sooner than elementary teachers and that teachers of chemistry and physics tend to leave sooner than teachers of other subjects. Any model that purports to offer subjectspecific supply and demand projections must take these subjectarea differences in attrition into account. Applying an undifferentiated average attrition rate to all categories of teachers is unsatisfactory. Incidentally, one of the unexpected findings of the recent attrition research, in light of concerns about possible shortages of mathematics and science teachers, is that mathematics teachers are not among those with shorterthanaverage lengths of stay in teaching. Moreover, even within the science category, the attrition patterns of biology teachers are quite different from those of chemistry and physics teachers. It appears, therefore, that "mathematics and science" is not a good category to use in assessing subjectspecific attrition.
Third, male and female attrition patterns differ. Women appear to have higher attrition rates early in their careers, but male and female rates converge later on (Grissmer and Kirby, 1987, 1991). There is also evidence of an agegender interaction effect, wherein the early attrition rates of young women are higher than those of young men, but attrition patterns of older male and female entrants are essentially the same (Murnane et al., 1988, 1989; Murnane and Olsen, 1989). Other things being equal, jurisdictions that hire different mixes of male and female teachers are likely to experience different rates of attrition—a finding that has not yet been reflected in supply projection models.
Fourth, it appears that attrition patterns may also differ significantly by race, although this is not as well established as the findings about other personal attributes. Racial differences would be important because of some policy makers' concerns about the future racial balance of the teaching force, and so there could be reason to bring this variable, too, into the attrition models.
Projection modelers up to now have produced disaggregated attrition
rates simply by computing separate attrition rates for the teachers in each subcategory, but as more teacher attributes are taken into account, the proliferation of subcategories makes this method infeasible. Analyses that go much beyond breakdowns by age and subject need to use multivariate methods to sort out the relationships of the different teacher characteristics to attrition. Multivariate methods are also better suited for estimating the effects on attrition rates of hypothetical or projected changes in particular teacher attributes. For example, the high early attrition rates of chemistry teachers may be due partly to some aspect of the field of chemistry as such (e.g., attractive employment opportunities outside teaching) and partly to age and gender differences between chemistry teachers and teachers in other fields. With a multivariate model, one can hold constant the effects of age and gender in order to estimate the marginal effect of subject specialty. Moreover, it is only with multivariate models that one can progress from mechanical projection models to behavioral models that allow, for example, for the effects of salaries, working conditions, and other policy variables on teacher attrition.
Variations in Attrition Rates Over Time
Even if all the difficulties of measuring and disaggregating attrition rates were overcome, the problem would remain of projecting attrition rates into the future. As will be seen, the prevailing approach in recent national and state models is to assume that attrition rates in future years will be the same as in the recent past. More specifically, the standard practice is to assume that attrition rates throughout the projection period (which may extend 10 or more years into the future) will remain fixed at the rates observed in the most recent year for which data are available. Yet ample evidence is now available that rates of teacher attrition, both in the aggregate and by subject field, have varied substantially in the past (Grissmer and Kirby, 1987: Coelen and Wilson, 1987) and, moreover, that such variations can be explained only partly by the changing age structure of the teaching force (Murnane and Olsen, 1990; Grissmer and Kirby, 1991). The assumption that agespecific, subjectspecific attrition rates will remain constant is therefore, to say the least, questionable, yet it is embedded in most of the recent projection models that have come to my attention. How one might project changes or trends in the rates is one of the major unresolved issues of attrition modeling.
The Quality of Retained Teachers
Finally, just as it would be desirable to project the demand for teacher quality, it would be valuable to be able to estimate the quality of future
retainees. A major concern of policy makers, unfortunately borne out by some research (Murnane et al., 1989; Murnane and Olsen, 1990), is that teachers' propensities to leave may be adversely selective in relation to quality—that is, better or "smarter" teachers are likely to leave sooner. Although the same problems of quality measurement arise in analyzing the quality dimension of attrition as in dealing with the quality dimension of demand, the modeling task is easier in the case of attrition because there is no need to estimate what the quality of teachers would have been under hypothetical conditions. Murnane and his associates, in the two studies just cited, have shown how a quality indicator (in their case, National Teacher Examination scores) can be included in an analysis of determinants of attrition along with other teacher characteristics. Thus far, however, no distinction by quality has been made, to my knowledge, in any attrition projection model. Introducing this element, in some state where test scores or other quality indicators are included in statewide teacher data files, would be a significant methodological advance.
National Projection Models and Attrition Estimates
In reviewing specific attrition models and studies, I distinguish first between explicit projection models and other research on attrition and second between nationallevel and statelevel analyses. The review is in three parts. This part deals with nationallevel projection models and attrition estimates; the following part covers statelevel projection models; and the third part examines selected studies of teacher attrition, which, although not intended to yield projections, do provide findings directly relevant to the development of projection methodology.
The NCES Projection Model
The NCES national projection model uses extrapolated nationalaverage attrition rates to project numbers of retained teachers in the future. The projection methodology has evolved but remains very simple, as the following comparison of the method circa 1985 and the method today will indicate.
Until recently, NCES estimated attrition by applying a single, nationalaverage, estimated attrition rate to all categories of teachers. It was assumed that this rate, set for many years at 6.0 percent, would remain constant over a projection period extending up to 12 years into the future (see, e.g., NCES, 1985). The estimates obtained by applying the 6.0 percent rate were said (in 1985) to represent "medium" projections. Alternative "high" and ''low" projections, corresponding to assumed attrition rates of 8.0 percent and 4.8 percent, were also provided. The 8.0 percent high estimate derived from a study of teacher turnover between 1968 and 1969 (Metz and
Fleischman, 1974). The 6.0 percent medium rate—frequently cited as the NCES estimate—was a downward revision of that number based on an NCES assessment (nature unknown) of altered market conditions (Cavin, 1986). The 4.8 percent figure was characterized as a theoretical minimum for reasons that I have not been able to discover. These rates, coupled with alternative projections of demand, yielded a wide range of estimates of the net demand for entering teachers during the projection period.
The present method (Gerald and Hussar, 1990) differs from the earlier method in several respects. First, the projections are now based on separate attrition rates for elementary teachers and secondary teachers rather than on a single average rate for elementary and secondary teachers combined. Second, the attrition rates that NCES now uses are derived from a broad study of separation rates by occupation in the U.S. economy conducted by the Bureau of Labor Statistics (BLS). ^{15} Third, NCES no longer assumes that attrition rates will remain constant in all future years but instead projects that they will follow a slowly rising trend. This trend projection comes from a rough model of retirement rates in the teaching force, based on age distribution data developed by BLS (for details, see Gerald and Hussar, 1990). Fourth, NCES still provides three sets of projections (low, middle, and high), based on alternative assumptions about how rapidly separations other than retirements will grow relative to projected retirements. According to the middle set of projections, the attrition rate for elementary teachers will rise from 5.5 percent in 1989 to 6.4 percent by the year 2001, while the secondary rate will rise from 6.9 to 8.0 percent over the same interval. The underlying BLS estimates of the separation rate, which are for 1983–84, are 4.9 and 5.6 percent for elementary and secondary teachers, respectively, employed full time in public schools. Apart from the elementarysecondary distinction, there is no differentiation of attrition rates by subject or by any teacher characteristic.
Although the present NCES attrition projection method is an improvement over the previous one, its limitations are evident. It yields only highly aggregative projections; it rests on a series of more or less arbitrary assumptions; and it depends on BLS attrition data collected seven years before the start of the projection period. However, it is pointless for two reasons to focus on these shortcomings today. One is that NCES has had very few options. Given the almost total lack of pertinent data, there is no way that a more sophisticated model could have been constructed. The other is that inquiry into the previous NCES approach has been rendered moot by the emergence of a major new data source, the NCES Schools and Staffing Survey (SASS). which will support much more detailed and sophisticated national attrition modeling than has previously been feasible. Although it is not yet clear precisely what form the new SASSbased models will take, I comment briefly on some possibilities below.
Attrition Estimates from the Schools and Staffing Survey
NCES has completed the first round (1987–88) of SASS data collection and processing and is now carrying out the second round (1990–91) of what is henceforth supposed to be a regular periodic survey. The 1987–88 SASS system collects data from linked samples of public and private teachers, administrators, schools, and LEAs. Also, of special significance for attrition modeling, it includes a followup survey, administered one year later (in 1988–89) to a subsample of the original sample of teachers, designed specifically to obtain information on patterns of teacher turnover.
The SASS surveys provide two types of data that can be used to estimate attrition rates. First, the public and private school surveys ask each school to report the number of teachers employed on October 1, 1986, who were no longer employed at that school on October 1, 1987, and then to indicate how many of the departed teachers were employed in another school at the latter date or engaged in various other enumerated activities. Second and more important, the SASS individualteacher surveys (the 1987–88 baseline survey and the 1988–89 followup) provide the wherewithal for a much more detailed analysis of attrition patterns and behavior. The baseline survey provides information on most of the personal characteristics of teachers that have been identified in previous research as major determinants of attrition. These include but are not limited to age, experience, gender, race, marital status, fulltime/parttime status, subject specialty, and salary. In addition, other SASS questionnaires provide data on schoollevel and districtlevel variables that may be associated with attrition, including pupil composition, locational factors, and certain working conditions. The teacher followup survey asks each respondent no longer employed at his or her original school to indicate, among many other things, whether he or she is still teaching and, if so, whether he or she is employed in another school in the same district, a school of a different district in the same state, or a school in a different state and if he or she has switched from a public to a private school or vice versa. Respondents no longer teaching are asked about their current activities. The followup sample is designed so that comparisons can be made among teachers who left the profession, teachers who switched schools, and teachers who remained at their original schools.
In an analysis of the schoollevel data, Rollefson (1990) estimated that the overall rate of attrition of public school teachers from the profession was 4.1 percent between fall 1986 and fall 1987 (4.2 percent among elementary teachers and 4.0 percent among secondary teachers). The estimated rates of attrition of private school teachers from the profession were much higher—9.3 percent for elementary teachers, 6.8 percent for secondary teachers, and 8.7 percent for elementary and secondary combined. The 4.2 and 4.0 percent estimates for public elementary and secondary teachers
are considerably lower than the BLS figures of 4.9 and 5.6 percent, respectively, that NCES has been using in its recent projection work. The difference may be partly due to timing, as the BLS figures are based on data for 1983–84.^{16} However, the fact that the SASS data yield an elementary attrition rate higher than the secondary rate is suspicious, considering that the opposite is true according to not only the BLS estimates but also most other attrition research.
The salient issue concerning the validity of Rollefson's figures is whether schools (i.e., school principals) are able to report with acceptable accuracy how many of their departing teachers have taken other teaching jobs. According to the SASS schoollevel data, the rate of attrition from the profession is less than half the rate of attrition from individual schools; that is, the majority of teachers who leave a particular school take jobs in other schools (Rollefson, 1990). Therefore, errors in reporting the leavers' destinations could easily distort the attrition rate estimates. An analysis of attrition rates based on the SASS individualteacher data should give some indication of whether these errors are substantial.
Just before this review was completed, NCES released a report (Bobbitt et al., 1991) containing initial attrition rate estimates based on the aforementioned SASS individual teacher baseline and followup surveys. According to this report, 5.6 percent of public school teachers left the profession between 1987–88 and 1988–89 (5.5 percent of public elementary teachers and 5.6 percent of public secondary teachers). The corresponding rates for private school teachers were 12.6 percent at the elementary level, 12.9 percent at the secondary level, and 12.7 percent for both levels combined. The report also confirms the Ushaped form of the ageattrition profile, shows a higher attrition rate for women than for men (5.8 percent versus 5.1 percent among public school teachers), and provides disaggregated estimates of attrition by teacher experience, degree level, race/ethnicity, and subject specialty.
The overall public school attrition rate of 5.6 percent reported in Bobbitt et al. (1991) is sharply higher than the 4.1 percent rate reported by Rollefson (1990). Part of the difference may reflect the different time periods to which the two sets of estimates pertain (i.e., the estimates based on the schoollevel data are for leavers between 1986–87 and 1987–88, while the estimates based on the individualteacher data are for the interval 1987–88 to 1988–89). Another part may stem from the fact that the 4.1 percent figure is for fulltime teachers only, whereas the 5.6 percent figure includes parttime teachers as well. It is also quite possible, of course, that the 4.1 percent figure is understated because of principals' inability to distinguish teachers leaving the profession from teachers transferring to other teaching jobs. A full analysis of the discrepancies has not yet been undertaken. The estimates based on the individualteacher data have the stronger a priori
claim to validity, provided that it is kept in mind that they reflect rates of attrition of fulltime and parttime teachers combined.
A more extensive attrition study based on the SASS individualteacher data is now being conducted under a contract from NCES by MPR Associates, Inc., of Berkeley, California. The MPR work will include not only tabulations of attrition rates, such as those produced by NCES, but also multivariate analysis, using discretechoice models, of influences on teachers' decisions to stay or leave. Results are expected in the latter half of 1992.
State Attrition Models
State models of teacher attrition (as of teacher supply and demand in general) vary widely in scope, detail, and analytical sophistication. The simplest attrition models merely apply an undifferentiated, statewideaverage, timeinvariant attrition rate to all categories of teachers. The more advanced models break down the attrition rate by age and subject specialty and apply the differentiated rates to numbers of teachers in each agesubject subcategory. The simpler models are of little methodological interest and are mentioned here only briefly. The bulk of this discussion focuses on the more elaborate methods.
Selected Simple Models
The simple, aggregative state attrition models are very similar to the models NCES has used to make national projections. A Nebraska study (Ostrander et al., 1988), for example, projects numbers of retainees by assuming a constant attrition rate of 9.0 percent for the period 1988–1994. An earlier California study (Cagampang et al., 1985) provides two alternative projections, one based on the assumption that the state's average attrition rate in the future will remain fixed at its baseyear value, the other based on a timetrend extrapolation of average attrition rates of the 10 most recent years. Neither of these studies projects numbers of retainees by subject area, and neither takes into account the effect on the attrition rate of the changing age composition of the teaching force.
Perhaps also worth mentioning under the heading of simple models is one that circumvents attrition projections altogether. A study of teacher supply and demand in Maryland (Maryland State Department of Education, 1988) deals with the supply of new entrants in some detail but bypasses an analysis of retention by relying on reports from local school districts on expected numbers of continuing teachers and vacancies. Setting aside the question of how well local districts can project their own numbers of retainees, it should be noted that this procedure seems to neglect the important point
that teachers who leave a particular district do not necessarily stop teaching in the state.
A model developed in South Carolina (South Carolina Department of Education, 1990) is one level more advanced than the simple models described above in that it breaks down attrition rates by subject area but not by age. In this model, state data files on individual teachers for the years 1985 and 1990 are used to determine net attrition rates during that fiveyear period for teachers in each of approximately 35 subject areas, and the resulting subjectspecific rates are then used to project numbers of retainees by subject area five years into the future—that is, to 1995. (Net attrition is defined as the percentage of teachers who leave less the percentage who return during the fiveyear period.) The implicit assumption is that the net attrition rate in each subject area will remain fixed during the fiveyear projection period at the average rate observed during the fiveyear base period. The state's data base, it turns out, contains information on every teacher in the state from 1981–82 to the present, which means that annual attrition estimates could have been developed; also, the data base would apparently support breakdowns by age and, presumably, other teacher characteristics.^{17} However, these features have not yet been incorporated into the South Carolina analysis.
Two other state studies, those for Ohio (Ohio Department of Education, 1988) and Wisconsin (Lauritzen and Friedman, 1991) produce subjectspecific projections of numbers of entering teachers (new hires) but avoid having to project attrition by instead projecting numbers of entrants directly. In the Ohio analysis, this is accomplished by making the assumption that the percentage of all teachers who are new hires in each subject area will follow its historical trend. That is, the model first projects teacher employment by subject area (demand) on the basis of projected enrollment and extrapolated teacherpupil ratios and then applies extrapolated newhire percentages to the results to obtain estimates of future numbers of entrants. In most cases, the extrapolations are produced by fitting simple linear regression equations to the historical data, but in some instances, the newhire percentages are estimated simply as averages of the actual percentages in a few recent years. Clearly, if numbers of new hires can be estimated independently, then projections of attrition are not needed to derive them. However, the Ohio method rests on the hardtodefend premise that the number of new hires in each field—which, after all, reflects not only attrition but also changes in enrollment patterns and staffing ratios—will remain in a stable relationship over time to total teacher employment in the field.
Similarly, in the Wisconsin analysis (Lauritzen and Friedman, 1991), future numbers of new hires are projected by extrapolating from numbers of entrants in the past. The Wisconsin approach differs from Ohio's, however, in that the projections are not based on a statistical trend extrapolation
method. Instead, the Wisconsin analysts apparently have assumed that certain ratios will remain constant over time—e.g., the ratio of the number of newly hired, inexperienced Wisconsintrained teachers in a particular subject specialty to the number trained in that specialty in the previous year—and have projected future new hires on the basis of past averages of such ratios. Unfortunately, the specifics of the projection methodology are too vaguely described in the Wisconsin report to permit a more detailed explanation here. In any event, the key point is that the Wisconsin study does not use explicit attrition estimates (in conjunction with demand estimates) to estimate numbers of new hires needed in future years.
The absence from the Wisconsin study of attrition projections and estimates of the demand for new hires based on them is surprising because the same group of analysts examined attrition patterns in some detail in earlier studies. For example, a report issued one year earlier by Lauritzen and Friedman (1990) presents not only attrition rates by certification category but also ageattrition profiles for particular subject fields. With such information, the Wisconsin group should have been able to produce the same kinds of detailed retention estimates as in the more advanced models that I describe below.
More Advanced Models
Three state models that I have had the opportunity to review—those for Massachusetts, New York, and Connecticut—project the supply of retained teachers on the basis of detailed attrition rates, disaggregated both by the teacher's age and the teacher's subject specialty or field of certification. I provide brief explanations of how each state model develops and uses the disaggregated attritionrate figures as well as comments on selected methodological points.
New York The New York teacher retention model (New York State Education Department, 1990) projects numbers of retained classroom teachers on the basis of separate retention rates for 15 subject areas and 7 age brackets: under 35, 35–39, 40–44, 45–49, 50–54, 55–59, and 60 and older. The retention rate for each agesubject subcategory is derived by comparing individual teacher records for the two most recent years for which data are available and determining the percentage of teachers present in the earlier year who were present in the later year as well. To project numbers of retained teachers, the New York analysts assume (implicitly) that the retention rate for teachers in each subject area and age bracket will remain constant at its baseyear value over a fiveyear projection period. Accordingly, they apply the aforesaid retention rates to the baseyear (1988–89) teaching force to project 1989–90 retainees; then apply the same rates to the
projected 1989–90 teaching force to project 1990–91 retainees; and so forth for years through 1993–94.
In addition to the attrition estimates themselves, a logically necessary element of such a model is a set of assumptions concerning the age distribution of newly hired teachers in each subject area. That is, in each year after the base year, the teaching force consists partly of retained teachers and partly of newly hired teachers, which means that information on the age distribution of the latter as well as the former is needed to project retention in subsequent years.^{18} The New York report says nothing about these assumptions. However, data tables provided to me separately by the New York State Department of Education include breakdowns of new hires by age for the base year, which suggests that the age distribution of newly hired teachers in years 1989–90 to 1992–93 was probably assumed to be the same as the distribution in 1988–89.
An important limiting feature of the New York model is the assumption that the attrition rate for each subject area and age will remain constant for the whole fiveyear projection period. (As will be seen, this assumption is not peculiar to the New York analysis but is shared by the Connecticut and Massachusetts models as well.) Several studies have shown that attrition rates vary substantially over time and have suggested that the rates are sensitive to market conditions prevailing in different periods (Grissmer and Kirby, 1987, 1991; Murnane and Olsen, 1990). With the multiyear teacher data files available to New York, it would be feasible to determine empirically whether the assumption of constant rates is warranted and, if not, to use some method—even if nothing more than trend extrapolation—to allow for changes in the agespecific, subjectspecific attrition rates over time.
A more specialized feature of the New York model that deserves scrutiny is that its age strata are defined in a way that masks the attrition rates of young, recently trained teachers. The lowest age stratum considered in the model, teachers younger than 35, includes, without differentiation, teachers in their first year of service and teachers with as much as 10 to 12 years of experience. Yet research has shown that attrition rates are much higher during the initial years of teaching than just a few years later (see, e.g., Grissmer and Kirby, 1987). Combining new teachers—those in the first 3 to 5 years of teaching—with teachers with 10 or more years of experience seems on its face to be an unpromising idea. The New York data base would certainly support finer disaggregation at the low end of the teacher age scale.
Connecticut The Connecticut model (Connecticut Board of Education, 1988) projects the supply of retained teachers on the basis of agespecific attrition rates for six subjectarea categories of classroom teachers plus two categories of staff other than classroom teachers. The classroomteacher categories are labeled (1) elementary, (2) special education and bilingual
education, (3) mathematics, chemistry, physics, and computer science, (4) teachers of required secondary subjects other than those in category 3 (i.e., teachers of English, social studies, biology, and earth science), (5) teachers of elective secondary subjects (foreign language, business and office practices, industrial arts, and other vocational fields), and (6) teachers described as ''K12 specials" (art, music, and physical education). The two nonteacher categories are "support staff, librarians, and media specialists," and "administration" (the latter including districtlevel as well as buildinglevel administrators and supervisors). Agespecific attrition rates within each category are computed for the base year 1986–87 by determining how many teachers employed at the beginning of the prior year did not return at the beginning of the 1986–87 school year.
The Connecticut model projects numbers of retained teachers for all school years through 2000–2001 by applying the aforesaid attrition rates, year by year, to data on the composition of the teaching force by category and age. (There is some ambiguity in the Connecticut report as to whether the attrition rates used for this purpose pertain to individual years of age or to fiveyear age brackets; however, the latter seems more likely because tables in the report present attrition rates for fiveyear intervals from 20–24 to 65–69.^{19}) The very strong implicit assumption underlying these projections is that the attrition rates of teachers in each staff category and age bracket will remain constant over the whole 14year projection period.
Assumptions about the age composition of newly hired teachers are particularly important in the Connecticut model because of the unusually long period for which projections are offered. At the average annual attrition rate of 5.8 percent cited in the Connecticut report, only about 43 percent of the initial teaching force would remain after 14 years, while the other 57 percent of the force would consist of teachers hired during the period. Therefore, estimates of the number of retained teachers, especially for the later years, depend strongly on whether new hires are assumed to be young, inexperienced teachers (presumably with high attrition rates) or older, more experienced teachers (presumably with lower attrition rates). According to the report, the Connecticut projections rest on the assumption that the age distribution of future entrants will be the same as that of entrants in the base year (1986–87). The report also refers, however, to an alternative "theoretical" age distribution for new hires derived from certain assumptions about future changes in the mix of sources of entering teachers. Exactly how this alternative distribution was derived is not explained, however, and, in any case, it does not seem to have been used in the projections. Nevertheless, the discussion of such a distribution does raise the important issue of whether analysts can do better than simply to assume that the composition of new hires will be the same in the future as in the past.
The Connecticut report also contains, in an appendix, a more detailed
analysis of attrition that takes into account gender and parttime status as well as age and subject and examines attrition patterns in 24 staff categories rather than the 8 described above. This analysis apparently uses multivariate techniques to sort out the effects of different teacher attributes on the propensity to leave teaching, but the specific multivariate statistical method is not explained. The extended analysis makes it possible, among other things, to distinguish differences in attrition rates associated with subject specialty per se from differences due to the varying age and gender composition of different fields. In addition, it demonstrates significant malefemale differences in attrition patterns, confirming a finding that has emerged in recent econometric studies of teacher attrition. Thus far, however, it appears that the results of this interesting part of the Connecticut work have not been reflected in supply projections.
The Connecticut model shares with the New York model the dubious assumption that all the agespecific, subjectspecific attrition rates will remain constant at their baseyear values over all years for which projections are made. If anything, that assumption is less tenable in the Connecticut case because of the unusually lengthy projection period. Connecticut, like New York, has multiyear data on individual teachers and has been in the business of calculating disaggregated attrition rates for some time. In fact. an earlier study (Prowda and Grissmer, 1986), generated the same types of agespecific, subjectareaspecific attrition rates for 1983–84 as are described here for 1986–87. The information would seem to be available, therefore, to determine whether the disaggregated attrition rates have changed systematically over time and, if so, to make some allowance for changes in the future.
A more specific question concerning the design of the Connecticut model is why teachers are grouped into only six categories and, especially, why into the six categories listed above. Some of the subjectarea groupings combine teacher categories that are (1) known to have different attrition rates and (2) unlikely to experience similar changes in demand in the future. For example, special education teachers are combined with bilingual teachers; physics and chemistry teachers are grouped together with mathematics teachers; teachers of biology and earth sciences are grouped with teachers of English and social studies: and foreign language teachers are combined with teachers of vocational education. Some realignment of these categories seems desirable. Moreover, the availability of a more detailed breakdown of teachers into 24 categories, as mentioned above, indicates that both realignment and further disaggregation are feasible.
Massachusetts The MISER model for Massachusetts (Coelen and Wilson, 1987, 1991a) disaggregates retention rates by age (fiveyear age brackets) and into 20 staff categories. The latter include 17 subjectarea categories plus administration, counseling, and media/librarian. The task of estimating
agespecific retention rates for each category appears to have been considerably more complicated for Massachusetts than for New York or Connecticut because of serious limitations of the available data files. Without attempting to go into detail, the basic problem seems to be that data on actual subjectarea assignments of individual teachers were not available, making it necessary to draw on teacher certification files and aggregate staffing data rather than teacher assignment data and to rely on various assumptions and imputations to disaggregate the teaching force by both age and subject.^{20} But in principle, if not in practice, the method of measuring retention is the same as in New York and Connecticut: annual retention rates are obtained by determining what percentage of teachers in each agesubject category employed in a given year are still employed in the following year.
The Massachusetts study differs from the New York and Connecticut analyses in that its attrition rate estimates are based on multiyear attrition data rather than only on data for a single base year but, unfortunately, the potential advantages of having multiyear data seem not to have been realized. Data on Massachusetts teachers were available for the 13 school years from 1973–74 to 1985–86, making it possible to compute 12 sets of yeartoyear retention rates. According to Coelen and Wilson (1991a), the retention rates actually used to project numbers of retainees in each staff category and age bracket are historical averages of the past rates. Based on further inquiry, however, it appears that the projections reflect only the last few years of subjectspecific attrition data and only the most recent age profiles of attrition. There is considerable ambiguity about exactly how the rates used in the projection models were derived (the MISER report itself is not forthcoming on this matter), but two points do seem clear: (1) the rates used in the projections do not reflect an analysis of how attrition rates are likely to change over time, and (2) the rates used are assumed to remain constant throughout the 10year period, 1986–87 to 1995–96, for which projections are offered.
The rationale for assuming that attrition rates will remain fixed in the future at their recent historical levels is, according to Coelen and Wilson (1991a), that there has been little variability in Massachusetts attrition rates over the years. It seems, however, that this contention is not well supported by the data. Both a table of retention rates by fiveyear age brackets (Coelen and Wilson, 1987:Table 12) and a set of timetrend regression equations (Table 14) show upward trends in attrition (declining rates of retention) during the period for which data are available.^{21} Therefore, unless rates stabilize or the historical trend reverses itself, the likely effect of basing projections on fixed historical rates will be to overestimate retention and consequently to underestimate the number of new teachers to be hired.
Like the New York and Connecticut models, the Massachusetts model also relies on the assumption that the age composition of new hires by
subject area will be the same throughout the projection period as in the base period. Although multiyear data on these age distributions are available, they are not used to explain or project trends (if any) in the ages of entrants. The Massachusetts model does go beyond the New York and Connecticut models, however, in that it breaks down new hires into three categories—reentering teachers, new entrants with certification, and new entrants hired under waivers of certification—and attributes separate age distributions to each. This makes sense because the new teachers and the reentering teachers are likely to have quite different age distributions. Whether this form of disaggregation improves the projections also depends, however, on whether the future mix of new and reentering teachers can be estimated satisfactorily—a question taken up in the following major section.
The MISER Multistate Model
As explained earlier, the analysts at MISER who developed the Massachusetts model are now engaged in a largerscale effort to construct the NEDSAD (Northeast Educator Supply and Demand) model for the New England states and New York. According to preliminary model specifications (Coelen and Wilson, 1991b), the treatment of teacher retention in this multistate analysis will be behavioral. Specifically, retention rates are to be estimated and projected as functions of teacher age, sex, and assignment; the female fertility rate by age (included because of a presumed relationship to the differential female rate of ''stopping out"); and the relative teacher salary by assignment (subject) category. Successful empirical implementation of such a model would represent an important step forward in the state of the art.
One aspect of the proposed model design is very disappointing, however, and seems to merit reconsideration. According to the preliminary model specifications (Coelen and Wilson, 1991b), only stateaggregate data on attrition rates within teacher categories (defined by age, sex, and subject specialty) will be used to model attrition behavior and then to project the future supply of retained teachers. If this is indeed to be the NEDSAD approach, a valuable opportunity will have been lost or deferred. The individualteacher data bases, assembled by MISER at great effort and expense, would support detailed and sophisticated attrition models, similar to those developed by Murnane and his associates and by Grissmer and Kirby at RAND (and which are discussed below). In fact, such models could deal with many more individualteacher, districtlevel, and statelevel influences on attrition than those mentioned in MISER's preliminary specifications. A model based only on aggregated data would be unlikely to yield satisfactory estimates of the effects of salaries and other causal factors on rates of attrition. Whether the potential benefits of the MISER effort will be real
ized depends to a considerable degree, therefore, on how this issue of level of aggregation is resolved.
Assessment of National and State Models
Although the foregoing review is not comprehensive, it does provide a reasonably good basis for assessing the present state of the art in projecting the supply of retained teachers at the national and state levels. In the following remarks, I confine myself to assessing the models on their own terms; that is, I take as given that the objective is limited to projecting numbers of retained teachers and refrain from criticizing models because they are nonbehavioral and do not deal with teacher quality.
The main attribute that now distinguishes relatively respectable attrition models from crude ones is disaggregation of attrition rates by age. Such disaggregation is essential to take into account the changes in overall attrition rates (in the aggregate and by field) that occur naturally, without any change in teacher or employer behavior, simply as a result of the changing age composition of the teaching force. Disaggregation by age requires data at the individual teacher level, but not all states with individualteacher files have made the transition to working with agespecific attrition rates. For instance, both Ohio and South Carolina seem to have the data needed to measure attrition by age, but both rely on average attrition rates (by subject area) to make projections. The most important step forward that they and other similarly situated states could take would be to shift to the types of disaggregated models developed for New York, Connecticut, and Massachusetts. States without statewide individual teacher data files would, of course, have to develop such data bases to be able to compete in the same league.
A serious limitation of the models that do disaggregate by age is adherence to the assumption that agespecific attrition rates will be the same in the future as in the period for which data are available. This assumption is not supported by the historical data, and there is little reason to believe that it will prove valid in the future. The developers of the New York, Connecticut, and Massachusetts models all had multiyear data to work with and could have tried to model changes in agespecific, subjectspecific rates over time, but the New York and Connecticut analyses do not draw on the multiyear data at all, and the Massachusetts analysis does not use the longitudinal information effectively. Even relatively simple trend extrapolations of the detailed attrition rates would seem preferable to the currently prevailing constantrate assumption.
The same criticism applies to the practice of assuming that the age distribution of new hires will remain fixed, even when multiyear data are available that could show whether and how those distributions have been changing. Moreover, the failure to distinguish (except in the Massachusetts
model) between new entrants and returning teachers in characterizing the age distribution of new hires is an easily correctable flaw.
There seem to be some major weaknesses in the methods used to disaggregate the supply of retained teachers by subject, but I say this hesitantly because these methods often go unexplained. One question is how teachers are categorized by subject for the purpose of modeling attrition. I suspect that they are often classified only by primary subject taught, which, if true, raises concerns about whether turnover rates are being measured correctly, especially in fields that teachers are likely to teach as sidelines. A related question is how teachers with multiple certifications are treated. Ideally, the projections would take into account the subjects that retained teachers are qualified to teach as well as the subjects they actually taught in the prior year, but no extant model, to my knowledge, has this capability. Consequently, the models may stimulate undue alarm about inadequate supply in particular subject areas. A third related question is how the models deal with teachers who transfer among fields. The logic of a subjectspecific model would seem to require treating each such teacher as a leaver from one field and an entrant into the other, but there is no indication in the reports I have reviewed that this is actually done. The usual approach, I suspect, is to categorize teachers only according to the subject they taught initially, and then to ask only whether they return in the following year, not whether they return to teach the same subject. If my suspicion is correct, both the projections of subjectspecific retainees and subjectspecific entrants may be distorted.
Finally, although the strong connection between age and attrition has been taken into account in some state projection models, there has been no such recognition of the relationships of attrition rates to other personal characteristics of teachers. In particular, none of the models that I have examined reflects the now wellestablished findings that attrition rates vary by gender and race and perhaps by marital status and fulltime or parttime employment. The Connecticut report (Connecticut Board of Education, 1988) is unique in that it includes a supplementary analysis of the relationship of attrition to age, gender, and fulltime/parttime status, but the results have not yet been used for projections. Bringing some of the additional factors into the models would be an improvement. However, to deal with additional variables would probably require switching from simply computing average attrition rates for different subcategories of teachers to inferring the rates from a multivariate statistical model. Such models have already been developed in research on attrition (see below). Applying them to supply projections is likely to be the next major advance in the state of the art.
Research on Patterns and Determinants of Attrition
Apart from efforts to project national and state supplies of retained teachers, major progress has been made during the last few years in research on the patterns and determinants of teacher attrition. The key to these efforts is the exploitation of largescale state data files, especially longitudinal files, on individual teachers. Using such data, researchers have been able both to characterize teacher attrition (and reentry) rates in considerable detail and to construct multivariate behavioral models that link attrition rates to economic and policy variables as well as to the teachers' own characteristics. The most important efforts along this line are those of Richard Murnane and his associates at Harvard and David Grissmer and his colleagues at the RAND Corporation, and it is mainly their work that I report on here.
The empirical research on teacher attrition has been conducted with two types of data bases: crosssectional and longitudinal. Each supports different types of analyses and different analytical methods. A crosssectional data set contains observations at a single point in time of all teachers, or a sample of teachers, teaching in a state or in the nation. A longitudinal data base provides repeated (timeseries) observations on members of particular cohorts of teachers—e.g., all teachers who began teaching in a state in 1975. The individuals represented in each cohort are homogeneous with respect to time of entry and, in some cases, age as well, and the longitudinal data make it possible to follow their careers over some period of years.^{22} It is possible, by collecting longitudinal data on enough different cohorts or, alternatively, collecting crosssectional data for many years, to have the best of both worlds—that is, to be able to carry out both crosssectional and longitudinal analyses with the same data base. Thus far, only an Indiana study by Grissmer and Kirby (1991), which has assembled 24 sets of annual observations on all teachers in the state, is in this enviable position.
CrossSectional Analyses
The disaggregated state modeling efforts described above are all crosssectional attrition analyses of a sort, in that they yield information on how teacher attrition rates vary with age and subject specialty, but they are very limited analyses in several respects. They do not take into account other factors associated with differences in attrition rates, do not examine the dynamics of attrition, and do not apply the multivariate tools needed to sort out the marginal effects of, and interactions among, the various determinants of attrition. Much more could be learned about attrition by analyzing in greater depth the data bases that some of these states have already used to project numbers of retained teachers.
The Connecticut report (Connecticut Board of Education, 1988) is somewhat more advanced than its counterparts in this regard because it supplements its projections with a more complete analysis of the determinants of attrition (found in an appendix and separate from the projection model). The main purpose of this analysis is to estimate the differences in attrition rates among subject areas that remain after controlling statistically for the age, gender, and fulltime or parttime status of the teachers in each field. Unfortunately, no details on the analytical method are provided—not even whether the multivariate analysis was conducted with individuallevel or aggregated data, so there is no way to judge whether the analysis provides a prototype potentially useful to other states.
The most important crosssectional attrition studies of which I am aware are the 1987 and 1991 RAND Corporation contributions by Grissmer and Kirby.^{23} The 1987 GrissmerKirby report, Teacher Attrition: The Uphill Climb to Staff the Nation's Schools, contributes to understanding teacher turnover by bringing together and comparing attrition data (mainly crosssectional) for several states and providing a detailed theoretical framework within which the empirical evidence can be interpreted. The study also offers a set of proposals for improving data on teacher attrition, some of which are now reflected in the NCES Schools and Staffing Survey.
The GrissmerKirby theoretical framework draws on theories of human capital, career progression, and imperfect information in the labor market. It explains, among other things:

The many factors that produce Ushaped ageattrition profiles—high attrition rates for the youngest and oldest teachers and much lower attrition rates for those in between:

Why the Ushaped curves are likely to have different shapes for men and women, with higher attrition rates for women in the early years of teaching;

How differences in opportunities outside teaching (opportunity costs) for teachers trained in different disciplines lead to differences in attrition rates among subject areas:

Why the influence of salaries and working conditions on attrition is likely to diminish as teachers progress in their teaching careers; and

How changing economic and demographic conditions may cause the whole attrition profile (i.e., the Ushaped ageattrition curves) to be higher in some periods than in others.
Grissmer and Kirby (1987) use crosssectional data for six states (crosssections for multiple years in most instances) to support their theoretical hypotheses. By comparing the Ushaped ageattrition profiles for different groups, they are able to confirm that attrition patterns differ for men and women, for elementary and secondary teachers, and for teachers specializ
ing in different subject areas. Then, using timetrend data from several states, they are able to demonstrate that overall attrition rates declined substantially from the 1960s and 1970s to the 1980s. They also present evidence that these declines are explainable only in part by the changing age composition (i.e., the aging) of the teaching force during the period in question and that attrition rates declined as well among teachers within specific age groups.
The 1987 GrissmerKirby analysis is not statistically elaborate. Its empirical contribution consists mainly of presenting and comparing ageattrition profiles. It offers no multivariate analysis of attrition, much less any behavioral model. Nevertheless, it has played an important role as catalyst and precursor by focusing attention on two central aspects of attrition analysis: (1) the underlying demographic determinants of attrition and (2) the dramatic but as yet inadequately explained changes in attrition patterns that have occurred over the years.
In their new study of Indiana, Grissmer and Kirby (1991) are able to explore the same and other aspects of attrition with a larger and richer data base—one containing data on all teachers employed in that state (about 50,000 at any one time) in each of 24 years. The crosssectional, descriptive portion of this analysis reconfirms that ageattrition profiles conform to the classic Ushaped pattern, but, in addition, the availability of 24 years of data makes it possible to examine in detail how the level and shape of these Ushaped curves vary over time. Grissmer and Kirby are able to show a continuing decline in these curves from 1965 to 1987. They also report that the dynamics of attrition have been different for women and men: the level of the attrition curve for women has declined more rapidly than the level of the curve for men, to the extent that male and female departure rates nearly converged by the mid1980s. These findings reinforce the conclusion that using a fixed set of attrition rates to project numbers of retained teachers 5 or 10 years into the future is not a satisfactory procedure. The most important contribution of the GrissmerKirby work on Indiana is not this crosssectional analysis, however, but rather a multivariate analysis of longitudinal data on entering cohorts, described separately below.
The only multivariate attrition analysis based on crosssectional data that has come to my attention is a study of teachers in Washington State by Theobald (1990). This study uses three sets of annual data on all certificated teachers in Washington (for school years 1984–85, 1985–86, and 1986–87) to investigate variations in, and determinants of, the rates at which teachers left their districts (i.e., the study is of districtlevel rather than statelevel attrition). Theobald presents a multivariate, discretechoice model in which each teacher's decision to leave or stay is represented as a function of (1) an array of individual teacher characteristics, including age, sex, race, years of experience, degree level, assignment (elementary or secondary),
and, notably, the teacher's projected nextyear salary and (2) a set of school district characteristics, including racial/ethnic composition of the student body, staff/pupil ratio, perpupil spending, perpupil tax base (assessed property value), enrollment, and the county unemployment rate (the last taken as an indicator of employment opportunities outside teaching). Using a maximumlikelihood probit estimation procedure, Theobald is able to derive estimates of the degree to which each of the aforesaid variables adds to or subtracts from the probability that a teacher will leave his or her district in a given year.
Theobald's findings are compatible in most respects with those of the GrissmerKirby studies (and of the longitudinal analyses reported later). Theobald captures the Ushaped form of the ageattrition relationship by representing attrition as a quadratic function of age. He shows that the attrition curves differ for men and women and that age and gender interact—that is, attrition rates are higher for young women than for young men but lower for older women than for older men. He also finds that teachers with higher degrees (men) tend to leave sooner than teachers without them (although some of the former "leave" by becoming administrators). Of special interest, he demonstrates that highersalaried teachers (again, especially men) are more likely to stay. He also shows influences on attrition rates of both district wealth and staff/pupil ratios, although the proper interpretations of these results seem to be uncertain.
Apart from the particular findings, Theobald's work is significant for demonstrating that a useful analysis of attrition patterns can be conducted with crosssectional data. This demonstration is important in connection with nationallevel modeling based on the SASS teacher survey data. The previously mentioned multivariate analysis of the SASS data by MPR Associates, Inc. is likely to rely on techniques similar to those used by Theobald (logit or probit modeling with multilevel data), and thus the Theobald work may provide a helpful prototype.
Longitudinal Analyses of Entering Teacher Cohorts
By following cohorts of entering teachers over time, it becomes possible to move from snapshot studies of attrition during a particular year to longitudinal analysis of teacher career paths. With longitudinal data, one can focus on the duration of each entering teacher's employment and on the factors associated with differences in length of stay. Moreover, one can consider not only whether teachers leave teaching at some point in their careers but also whether they return later (and, if so, when) and how long they remain in the teaching force the second time, or subsequent times, around. The combination of longitudinal data bases with the recently developed statistical techniques of survival and hazard modeling (see below)
creates favorable conditions for developing useful behavioral models of the supply of retained teachers.
In a series of recent articles, Richard Murnane and his colleagues have analyzed the career paths of entering teacher cohorts in detail, considering both attrition after initial entry into teaching and the likelihood of commencing and continuing subsequent teaching spells. Working primarily with data from Michigan and North Carolina, they have been able to relate teacher entry, persistence, and exit behaviors not only to the teachers' own characteristics but also to salaries, opportunity costs, and various school district characteristics. Although the findings in their present form are not directly usable for projections, this line of work could eventually lead to a new class of behavioral projection models.
In a study of Michigan public school teachers who entered teaching in 1972 or 1973, Murnane et al. (1988) use the statistical technique of proportional hazards modeling to relate the length of stay in teaching to the teacher's age, sex, and subject specialty. They find, as expected, that younger teachers leave sooner than older teachers and women leave sooner than men; but they also report a strong interaction effect between age and sex—namely, that the malefemale difference in the attrition rate exists for younger (under 30) entrants only. They also find important differences in attrition patterns by subject field.^{24} Controlling for age and gender, teachers of chemistry and physics are likely to leave teaching the soonest (i.e., their attrition rates in the early years are the highest); teachers of English are the next most likely to leave; and teachers of other subjects tend to remain longer. Interestingly, teachers of mathematics, often lumped together with science teachers in discussions of ''shortage," are not among those with particularly high attrition rates—a finding that has since been corroborated by other studies.
In a fartherreaching study of entering Michigan teachers (entrants between 1972 and 1975) Murnane and Olsen (1989) examine the relationship of attrition not only to age, gender, and subject specialty but also to salaries in teaching, salaries in occupations outside teaching, and a variety of school district characteristics. The inclusion of these additional variables puts the analysis into a different class from the analyses described previously, as it introduces the possibility of linking attrition to economic conditions and policy variables as well as to characteristics of the teachers themselves. In this study, Murnane and Olsen use an analytical method known as waiting time regression to relate the duration of each teacher's stay in teaching (and hence, the rate of attrition) to the aforesaid set of explanatory variables. In addition to confirming the same general age, gender, and subjectfield effects on attrition as in the Michigan study cited just above, they show that (1) higherpaid teachers are likely to remain in teaching longer and (2) teachers prepared in fields that offer higher salaries outside teaching (espe
cially chemistry and physics) are likely to leave sooner. These findings support the views that teacher supply is sensitive to the relative salary level in teaching and that salary movements can play the same role in equilibrating supply and demand in teaching as they do in labor markets generally.
Murnane and his associates have also worked extensively with data on entering teacher cohorts in North Carolina. In Murnane et al. (1989), hazard models are used to analyze attrition/retention patterns among individuals in that state who entered teaching in 1976 and 1978.^{25} These models, which are constructed separately for elementary and secondary teachers, relate each teacher's length of stay to age, gender, the agegender interaction, subject specialty, selected district characteristics, salary, and—a unique feature—the teacher's score on the National Teacher Examination (NTE). As in the Michigan study, attrition is higher in the initial years of teaching than in later years; there are significant differences in median duration among subject specialties; and higher salaries are associated with higher retention rates, but with the influence of salary on retention diminishing over time. In addition, the North Carolina analysis yields the striking finding that teachers who score higher on the NTE tend to leave sooner—that is, higherquality teachers (to the extent that NTE score signifies quality) are less likely to remain in the retained teaching force. ^{26} This demonstration, in a multivariate framework, that NTE scores are negatively associated with retention rates appears to be the closest that anyone has yet come to dealing with teacher quality in an analysis of teacher supply.
In another North Carolina study, Murnane and Olsen (1990) use the aforementioned waitingtime regression technique to analyze attrition patterns among teachers who began their careers in the years 1979 to 1984. The principal findings, this time reflecting data on a larger number of entering cohorts, reinforce those of the study mentioned just above: (1) there are important differences in expected duration among subject areas, with teachers of chemistry and physics likely to leave soonest, teachers of English likely to leave somewhat later, and elementary teachers and—again—mathematics teachers likely to remain the longest; (2) betterpaid teachers are likely to remain in the profession longer than lowerpaid teachers: and (3) teachers with relatively high NTE scores are likely to leave sooner than their lowerscoring colleagues. In addition, this study demonstrates that attrition rates depend on certain characteristics of local school districts, such as size, income, and socioeconomic composition, as well as on characteristics of the teachers themselves. Differences in these characteristics. according to the authors, may serve as proxies for differences in teachers' working conditions, which, in theory, should affect attrition in much the same way as differences in salaries. Finally, the study demonstrates that intercohort differences in attrition patterns persist even when all the other
factors mentioned above are held constant, confirming that the standard projectionmodel assumption of constant rates is unfounded.
The Grissmer and Kirby (1991) study of Indiana, described earlier, offers two different longitudinal analyses of the attrition behavior of entering cohorts. In the first analysis, all individuals who began teaching in years 1965–66 to 1982–83 (18 successive cohorts) are followed through the year 1988. Using a proportional hazards model, Grissmer and Kirby relate each individual's duration in teaching to the individual's age, gender, subject specialty, and year of entry. The findings fit closely with others reported above: the age profiles have the usual shape; similar malefemale differences appear: and the differences in attrition patterns among subject areas resemble those found in other states. In addition, year of entry clearly matters, reinforcing the point that it is inappropriate to assume constancy of attrition rates over time.
In their second analysis, Grissmer and Kirby again use proportional hazards modeling but, in this instance, they focus exclusively on attrition during the first four years of teaching among those who entered the profession between 1965–66 and 1985–86 (21 entering cohorts). The models. estimated separately for men and women, relate duration in teaching not only to age and subject specialty but also to each individual's starting salary, the teacherpupil ratio in the individual's district, and, as options, the general labor force participation rate in the economy and the median income of fulltime workers. The rationales for the last two factors are, respectively, that the labor force participation rate is an indicator of alternative economic opportunities, especially for women, and that the median income of workers is a proxy for the level of compensation outside teaching. With these models, Grissmer and Kirby are able to show that early attrition rates are (1) highly sensitive to salary, corroborating the findings of Murnane's Michigan and North Carolina studies, and (2) sensitive to working conditions as well, insofar as these are represented by the teacherpupil ratio. They also find that declining female attrition rates are associated with the increasing labor market participation rate for women. They are not able, however, to demonstrate an effect of earnings outside teaching, possibly because their median income variable is too crude an indicator of the opportunity wage.
Assessment and Implications
Teacher attrition is a field in which research seems to be playing just the right roles in relation to practice—opening up new issues, generating new knowledge, and yielding both analytical methods and substantive findings that can be put to practical use. The attrition studies described above
have already produced important information that we can expect to see reflected, sooner rather than later, in state and national models for projecting the supply of retained teachers. Moreover, this line of research is relatively new, and its potential has only begun to be explored; if it continues, more practical applications are likely to follow. In particular, the possibility seems well within reach of developing a new class of researchbased, behavioral, attrition projection models.
Among the potential direct and immediate contributions of the research to projection modeling, I would cite the following:

Findings about the Ushaped ageattrition profiles are so compelling that they should convince any state or local model developer who does not already disaggregate attrition rates by age (and who has individualteacher data) that such disaggregation is essential.

Research findings concerning attritionrate differences among categories of teachers should guide model developers in disaggregating their projections by subject specialty and grade level.

Findings about malefemale differences in attrition patterns, which are not currently reflected in projection models, can easily be incorporated into models based on individualteacher data.

The strong indications from the research that attrition rates vary over time should help convince model developers to abandon the assumption that rates will remain constant and to introduce timevarying rates into their projections, even if, initially, these are based on nothing more than trend extrapolations.
I would place in a different category research findings pertaining to the effects on attrition of such factors as teacher salaries, opportunity wages outside teaching, working conditions, and other characteristics of schools and school systems. Although the findings to date demonstrate that such factors influence attrition, the model specifications and measures on which they are based are not yet well enough developed to yield good estimates of the magnitudes of the effects. As examples, measures of opportunity wages are crude; only a few rough indicators of working conditions have been tested (and those may not be measured at the appropriate level of aggregation); and the effects of such other factors as pupil composition, size of school, and type of community have not been examined systematically. Also, models that allow for both timevarying factors and timevarying effects of each factor apparently have not yet been applied to the attrition problem. Thus, although the relevance of behavioral attrition models has been established, considerably more work on model design will be needed before such models can be developed for practical use.
MODELS OF THE SUPPLY OF ENTERING TEACHERS AND THE SUPPLYDEMAND BALANCE
According to the conceptual scheme underlying, or implicit in, most efforts to model teacher supply and demand, projections of the supply of entering teachers should provide the third and final ingredient needed to assess the future supplydemand balance. The two model components discussed in previous sections, projections of demand and projections of numbers of retained teachers, taken together, yield estimates of future demands for new hires and comparisons between the projected numbers of new hires and the projected supply of entrants should lead to conclusions about the adequacy of supply and the likelihood of future shortages or surpluses. This implicit modeling strategy is only slightly closer to accomplishment today, however, than it was five years ago—the principal reason being that projecting the supply of entrants has remained an intractable problem. I would adhere today to the conclusions I expressed in my earlier review (Barro, 1986):
Projecting the supply of [entering] teachers, and hence the total teacher supply, is beyond the capacity of all the current models. The methods that have been proposed for making such projections have fatal conceptual flaws, and the results—though perhaps useful for some purposes—cannot legitimately be labeled supply projections. Little credence should be given, therefore, to any findings of ''shortage" or "surplus" based on comparisons between projections of demand and purported projections of supply.
This negative general conclusion notwithstanding, there is some positive news to report. Some modest steps toward improved teacher supply modeling have been taken in recent years. More state studies now take account of multiple sources of teacher supply, including the supply of returning former teachers and other hires from the "reserve pool"; fewer focus solely on the flow of new graduates from teacher training institutions, as most studies did in the past. There seems to be heightened awareness of key conceptual problems in modeling the supply of entrants—for instance, that past rates of entry into teaching were determined more by demand than by supply—even though methods for dealing satisfactorily with these problems have not yet been developed. Perhaps most encouraging, researchers have begun to deal analytically and behaviorally with certain pieces of the supply problem and to create the largescale data bases needed to support further analytical efforts. Thus, although the basic problem remains unsolved, it does not follow that nothing worthwhile has been accomplished.
Considerations in Modeling the Supply of Entrants
Given the present undeveloped state of models for projecting the supply of entering teachers, it is at least as important to focus on conceptual issues
and general approaches as on specific modeling techniques. The following are some of the key issues and considerations.
Definitions and Concepts of Supply
The supply of entering teachers in a state or in the nation is the number of persons (not already teaching) who are available and willing to take teaching jobs. Like the demand for teachers, the supply of teachers is contingent on multiple factors, including the standards that an individual must meet to be deemed eligible to teach and the salaries, benefits, and working conditions that prospective employers are offering. Thus, it is meaningful to speak of the current or projected supply of entrants under currently prevailing conditions (current certification standards, salary schedules, etc.) and the altered levels of supply that would correspond to alternative sets of conditions. Ideally, a projection model would yield a mathematical relationship between the supply of entrants and all the aforesaid factors. Thus far, however, we have yet to achieve even the more modest goal of projecting supply under the assumption that all pertinent factors will remain fixed at their current values.
The key conceptual problem in projecting supply (or even in measuring the current supply) is inherent in defining supply in terms of the availability and willingness of eligible persons to take teaching jobs. Availability and willingness are unobservable mental states. What one does observe with the personnel data normally obtainable are the numbers of persons actually hired as entering teachers, not the numbers who would have taken positions had they been offered. The number of persons willing to supply their services as teachers in a state might be 10 percent, 50 percent, 100 percent, or 500 percent greater than the number actually hired; there is no way to determine the supply of wouldbe entrants from employment or hiring data alone.
But, one might object, the same holds true of demand. The demand for teachers is defined as the number that school systems are able and willing to pay for (of specified quality, at specified salaries, etc.), but all one can observe is how many teachers districts actually employ, not the possibly larger number that they might employ if more qualified applicants were available. This apparent symmetry is misleading, however. Generally speaking, the teacher market in the nation and in the states, albeit with local and subjectspecific exceptions, is characterized by excess supply—more trained and eligible applicants than there are teaching jobs. States and school systems, for the most part, can obtain the aggregate numbers of teachers (if not the precise characteristics of teachers) that they want. If they attract too few applicants, they can relax their hiring standards. In other words, setting aside the quality issue, the number of teachers actually employed closely
approximates the number demanded. In contrast, not all applicants for teaching jobs are able to obtain them. The number of teachers employed is smaller than total teacher supply, and the number of entering teachers hired each year usually falls short of the number seeking teaching positions.
The implication for projection modeling is that, although one can approximate future demand by projecting employment, one cannot estimate the future supply of entrants by projecting the number of teachers hired. The number hired ordinarily reflects only a fraction of the supply—namely, teachers actually selected for employment. Consequently, no analog of the previously described mechanical approach to projecting demand and attrition is available for projecting the supply of wouldbe entrants into teaching. The supply of entering teachers must be inferred indirectly rather than merely extrapolated, but such inference requires methods more sophisticated and demanding of data than any currently employed.
Faced with uncertainty about what one should project under the heading "supply of entering teachers," model developers have generally fallen back on one or the other of two unsatisfactory conceptions of supply. One notion is that the future supply of potential entrants into teaching can be represented by projecting the number of persons with teaching certificates and/or the annual increment therein. The other is that the supply of entrants can be projected by applying projected entry rates, based on actual entry rates observed in the past, to the numbers of persons in various pools from which teachers are drawn. Neither is logically sound. The details are spelled out later. Suffice it to say that the first approach, projecting future numbers of certificated persons, falls short because certification is only tenuously linked to supply, while the second approach, projecting numbers of entrants, fails because entry rates, both past and future, are more likely to be determined by how many teachers school districts want than by how many are in the supply. The upshot, therefore, is that there have been no valid projections of the supply of entrants and, consequently, no meaningful assessments of the supplydemand balance.
Sources and Categories of Entrants
Apart from the underlying conceptual problems discussed above, an important complicating factor in modeling the supply of entrants is that entering teachers come in different categories and from multiple sources. The propensities of individuals to enter teaching vary greatly among these categories, as do the factors that influence supply behavior. Instead of focusing on a single supply projection model, therefore, it is more realistic to think in terms of a set of submodels, each pertaining to supply from a different class of potential teachers.
There is no one generally accepted taxonomy of sources or categories.
The most commonly encountered distinction is that between the flow of newly produced teachers (recent graduates of teacher training programs and/or persons recently certificated to teach) and the flow of teachers from the socalled reserve pool. This important concept, the reserve pool, is defined narrowly by some analysts and broadly by others. According to the most common narrow definition, the reserve pool is the stock of persons not currently teaching but certificated to teach in a state. More broadly conceived, the reserve pool encompasses all persons eligible to become teachers or even, in some versions, a state's (or the nation's) entire collegeeducated population. It has become apparent, however, that the twoway distinction between newly produced teachers and reservepool members is inadequate for modeling supply. For instance, the supply behavior of teachers who have recently left teaching jobs is very different from that of certificated persons who have never taught, even though both groups are considered to be in the reserve pool. In practice, most state studies of the supply of teachers to the public schools classify potential entrants into finer categories, based on various combinations of the following distinctions:

Entrants with and without previous teaching experience (i.e., new or entrylevel teachers versus experienced teachers). Note that the latter may or may not be defined to include teachers with experience in private schools.

Newly trained teachers (new graduates of teacher training programs) versus teachers who graduated in earlier years,

Teachers who have never taught in the state versus teachers previously employed in the same state (the latter usually described as reentering teachers),

Teachers resident in the state versus immigrants from other states,

Teachers trained in a state's own teacher training institutions versus teachers trained out of state, and

Teachers with and without full certification (or with different levels of certification, where applicable).
Different model developers seem to have come to different conclusions about which sources and categories need to be analyzed separately in projecting the supply of entering teachers. For example, the basic distinction in the Connecticut analysis (Connecticut Board of Education, 1988) is between firsttime certificated persons and certificated persons in the reserve pool: the main distinction in the Massachusetts model (Coelen and Wilson, 1987) is between firsttime entrants (persons who have not previously taught in the state) and reentrants; and the two key distinctions in the Maryland study (Maryland State Department of Education, 1988) are between instate versus outofstate and experienced versus inexperienced teachers.
Disaggregation by Subject Specialty
To analyze the supplydemand balance by field of teaching, one needs disaggregated projections of the supply of entrants as well as of teacher demand and the supply of retainees. This need to disaggregate complicates the supply models, just as it complicates models of attrition, and raises certain conceptual problems that appear not to have been addressed in the modeling literature.
One issue that arises is how one should deal with potential entrants who hold certificates in multiple fields. If an applicant is certificated as both a physics teacher and a mathematics teacher, for example, should that person be counted in the supply of potential entrants to both fields, or should he or she only be counted once (if so, in which field?) or somehow prorated between fields. How multiply credentialed teachers are treated obviously can affect subjectspecific assessments of the supplydemand balance. Much the same issue was raised earlier in connection with the supply of retained teachers, but the appropriate answer is more elusive in the case of entrants because retained teachers can be categorized according to the subjects they have actually been teaching, but potential new teachers cannot.
Although the classification issue may seem like a matter of bookkeeping, it can have substantive implications—that is, it can affect supply projections and the subsequent assessments of supplydemand balance. An arbitrary method of fitting each prospective teacher into a single subject field, for example, or of assigning hypothetical fractions of such individuals to different fields, can generate findings of shortage where none actually exists. It seems important not to make the teacher supply seem less flexible than it is, which means that the capability of teachers and prospective teachers to teach multiple subjects should not be ignored. Thus far, however, none of the models I have seen takes the possibilities of substitution and mobility across fields adequately into account. How these possibilities can be accommodated is an interesting unsolved problem in model design.
Disaggregation by subject also creates a new category of entrant not mentioned in the foregoing taxonomy of supply categories and sources—namely, entrants into a particular subject area who are transferring from another subject area in the same state or school system. Because such persons are entrants in one sense but retained teachers in another, it would seem unreasonable to lump them together with other entrants for the purpose of modeling or projecting subjectspecific supply. Yet they must be taken into account somehow to make assessments of the supplydemand balance come out correctly. The handling of such transferees is currently one of the loose ends in supplydemand modeling. It is not clear how such persons are counted or whether they are included in entry rates or, if so,
whether they are treated as new or experienced teachers. Certainly, no theory has been offered of the supply behavior underlying transfer decisions.
The Quality Dimension of the Supply of Entrants
Changing the quality standards applicable to entrants is the principal strategy available to a school system or state for eventually altering the quality of its teaching force. Both the feasibility of improving quality and the rate at which quality can be improved depend on the distribution of quality in the supply of prospective new hires. In principle, and assuming the availability of suitable quality indicators, prospective entrants (both newly trained teachers and members of the reserve pool) could be classified by quality, and separate entry rates or propensities to apply for or accept positions could be determined for persons in the different quality strata. Alternatively, quality could be taken into account in a multivariate analysis of determinants of entry into teaching. The latter approach has, in fact, been implemented in a recent study of new teachers in North Carolina (Murnane and Schwinden, 1989), but the results are not directly transferable, for reasons to be explained, to teacher supply projection models. At present, the unsolved basic conceptual problems of supply modeling are so severe that taking the quality dimension into account seems at best a remote prospect.
State Projection Models
Given what has been said thus far about the conceptual obstacles to projecting the supply of entrants, it should not be surprising that the socalled supply projections presented in state supplydemand studies are, almost invariably, something other than true projections of supply. Mostly, these projections are of the two types mentioned earlier: (1) estimates of the future flow of persons into certificated status and/or the stock of certificate holders or (2) estimates of future numbers of entrants into teaching of various types and/or from various sources. Such projections are not necessarily uninformative. They may, in fact, be of considerable value either for reassuring policy makers that future supplies of teachers will be adequate or for identifying areas in which recruitment may become difficult. But neither type answers the question that one would hope to have answered by a supply projection: How many teachers (of specified qualifications or quality) are likely to be available and willing (under specified conditions) to enter the teaching force in the future?
The following descriptions of selected state models cover both the methods used to project the supply of entering teachers and the corresponding approaches (if any) used to assess the supplydemand balance.^{27} The sequence
of descriptions is arranged to emphasize the different concepts and modeling strategies now in use. An overall assessment of the models is provided at the end.
Projections of Numbers of Certificated Teachers
Two models that I have examined, those for Nebraska and South Carolina, reflect the notion that the stock or flow of persons with teaching certificates constitutes a state's teacher supply.
Nebraska The Nebraska analysis (Ostrander et al., 1988) equates supply (of new entrants) to the number of persons added annually to the ranks of Nebraska certificated teachers. In principle, the analysis recognizes three sources of potential new hires: (1) newly certificated Nebraska graduates, (2) outofstate teachers who become eligible to teach in the state, and (3) persons in the reserve pool—defined as the stock of persons holding certificates but not currently teaching. Entrants from the reserve pool are excluded from the supply projections, however, because of lack of information on how many reserve pool members might actually be available to teach. The projection method applied to the two newflow categories is simple trend extrapolation. The number of new graduates from Nebraska teacher training programs is projected to follow its historical downward trend (a decline of about 3 percent per year); the same trend is assumed for new graduates in fields other than teacher training who obtain teaching certificates; and the number of newly certificated teachers from other states is also extrapolated on the basis of a continued declining trend (in this case, a 4 percent decline per year). No attention is given in this analysis to the rates at which newly certificated persons actually enter teaching, nor to the rate at which former teachers return (the latter is part of the unanalyzed supply from the reserve pool). Thus, the model is seriously incomplete. Nevertheless, the Nebraska report proceeds to compare the projected demand for new hires (projected demand less projected number of retainees) against projected supply of new certificants, as defined above, and, finding the latter greater than the former, declares that the state will be enjoying a teacher surplus.
South Carolina A model developed in South Carolina (South Carolina Department of Education, 1990) also equates the supply of entrants to the number of new certificates issued annually. A distinction is made between new certificate holders trained in South Carolina institutions and those trained outside the state (the latter accounting for almost half the newly certificated individuals in recent years), but the same projection method is applied to both categories. This method consists simply of assuming that the annual number of new certificate holders in each subject area will be the same in
the projection period (1990–1995) as it was, on average, during the fiveyear base period, 1985–1990.
The South Carolina report acknowledges the importance of returning teachers as a source of supply (30 percent of leavers return) but does not project their numbers explicitly. Instead, it deals with returning teachers on the demand side of the model by measuring and projecting attrition in terms of the net number of leavers (i.e., the number of leavers net of returnees rather than according to the usual concept of gross annual attrition. No mention is made of any hires of certificated persons from the reserve pool other than these returnees.
Based on this model, the South Carolina analysts characterize the future supplydemand balance in each subject area in terms of the annual projected gain or loss of teachers—that is, the difference between the projected number of new certificate holders and the projected demand for new hires. A loss is said to signify increasing future difficulty in finding enough teachers. This projectionbased assessment is supplemented by separate assessments based on districtreported shortages and numbers of vacant positions. Overall rankings of subject fields according to degree or severity of shortage are then constructed by combining rankings of losses based on the projections with rankings based on the survey data. Special education, science, foreign language, library science, and mathematics turn out to be the principal fields of shortage.
Projections of Entrants by Category
Two other state analyses, those for Maryland and Wisconsin, present relatively simple models based on the premise that one can estimate the future supply of entering teachers by projecting numbers of entrants or rates of entry from various pools of potential teachers. The MISER study of Massachusetts presents a more elaborate model founded on essentially the same proposition.
Maryland The Maryland model of teacher supply (Maryland State Department of Education, 1988) recognizes four categories of entrants: (1) instate returning experienced teachers (i.e., teachers formerly employed in Maryland schools), (2) beginning teachers who are graduates of Maryland teacher training programs, (3) beginning teachers who are graduates of outofstate teacher training programs, and (4) outofstate experienced teachers. The number of new Maryland graduates who will enter teaching in each subject area is projected by applying a projected entry rate to an estimate of the future number of graduates with that subject specialty who will emerge from the state's teacher training programs. The estimated numbers of graduates by field are compiled from data supplied by individual
Maryland institutions of higher education. The projected entry rates appear to be the actual rates in the most recent year for which data are available, but whether these rates are subjectspecific is not clear from the Maryland report. Numbers of entrants (by subject specialty) from the other three sources of supply are projected, it appears, by assuming either that they will be the same as in the most recent year for which data are available or that recent trends in the numbers will continue. No explanations are provided of the specific assumptions applicable to each category.
As to the supplydemand balance, the Maryland study reports a ''projected discrepancy" for each subject specialty—the difference, whether positive or negative, between the projected supply of and demand for entrants—and also the projected supply of entrants as a percentage of the projected demand. These percentages are as low as 60 to 70 percent for some subjects (referred to as fields of "critical shortage"), but the fact that such large shortfalls are anticipated seems not to be taken as cause for alarm. Instead, the report merely notes that Maryland school systems can expect to fill only some of their vacancies from "normal sources of supply" in the years for which the deficiencies are projected.
Wisconsin The Wisconsin report on teacher supply and demand (Lauritzen and Friedman, 1991) presents unusually detailed descriptive information on numbers and characteristics of entrants into teaching but only a rudimentary and incomplete model for projecting the entrant component of supply. The projection method consists, in essence, of calculating the number of new Wisconsintrained teachers who would be hired in each subject specialty if the same percentages of new graduates were hired in the future as in the recent past. The study does not explicitly project rates of entry from other sources (e.g., returning former teachers and teachers imported from other states) but seems to assume implicitly that such entrants will account for the same percentage of total entrants as in prior years. Lauritzen and Friedman give special attention to two phenomena neglected in many other state studies: (1) that the entry of newly trained teachers into the profession is distributed over time, not all concentrated in the year following certification, and (2) that expected entry rates for persons holding multiple certification are higher than for persons certificated in only a single subject specialty.^{28} In general, however, their study is more useful for (and oriented toward) assessing the employment prospects for persons trained, or in training, in different teaching fields than for projecting the future supplydemand balance in the state.
Massachusetts The MISER model for Massachusetts (Coelen and Wilson, 1987, 1991a), defines supply as, in essence, a weighted sum of numbers of potential teachers in multiple categories; the weights are historically determined probabilities of entry. The model recognizes three broad types of entering teachers: (1) reentering teachers—teachers who have taught
previously in the state but have interrupted their careers (also referred to as stopouts); (2) new entrants with certification—a category that mixes together (a) teachers who have just graduated and been certificated in Massachusetts, (b) persons certificated in earlier years but who have never taught in Massachusetts, and (c) certificated teachers immigrating from other states: and (3) entrants hired under waivers of certification.
The supply of reentering teachers is projected by applying historical transition rates to the numbers of teachers who left in each prior time period. These rates (probabilities of reentry) vary with the length of time since leaving active teaching, reflecting the wellestablished finding that recent leavers are more likely to return than those who left many years ago. The transition rates are derived from a regression procedure, in which numbers of reentrants in a given year are related to numbers of leavers in past years, but the specifics of the regression methodology are unclear. It does appear, however, that year of departure is the only attribute considered in estimating the probability that former teachers in the reserve pool will return to teaching; such other factors as the teacher's age, gender, and subject specialty are not taken into account.^{29}
The supply of new entrants with certification is also estimated by applying a set of estimated transition rates to numbers of persons in the pool of certificate holders who have never taught. In this case, the transition rates vary according to the time elapsed since certification was obtained. Individuals recently certificated are likely to enter in relatively large percentages; those who received their certification five or more years ago have much lower entry probabilities. Note that new graduates of teacher training programs—a category singled out for special attention in most other analyses—are here treated merely as certificate holders of the most recent vintage. Coelen and Wilson (1987) indicate that a regression model was used to estimate the set of transition rates, but the details of the modeling technique are not laid out. Again, it appears that no characteristics of certificated persons other than time since certification are taken into account in estimating the entry probabilities; that is, the projected entry rates are not differentiated by age, sex, or subject specialty.
The number of entrants in the final category, teachers hired under waivers of certification, is projected as a fixed percentage of the combined number of teachers hired in the previous two categories. The rationale for this specification is difficult to discern. It would seem more natural to treat the noncertificated group as a residual category, hired in such numbers as necessary to fill out the ranks of the teaching force. Such treatment would imply an inverse relationship rather than a proportional relationship between numbers of certificated and noncertificated entrants (holding the total number of new hires constant). It would be interesting to know which approach is more consistent with the empirical evidence.
Two other aspects of the treatment of the supply of entering teachers in the Massachusetts model are noteworthy. First, in addition to modeling rates of entry of certificated persons into active teaching, Coelen and Wilson also seek to explain the rate at which teachers obtain certification. They report what they characterize as a myopic pattern of behavior, whereby the number of new certificates issued in a year appears to depend mainly on the number of teachers hired in the immediately preceding year (Coelen and Wilson, 1991a). It is not clear how this pattern relates to the flow of potential new teachers through the training pipeline, nor whether it is largely explained by a phenomenon alluded to in the 1987 Massachusetts report—namely, that some teachers obtain certification at or after the time they are hired.
Second, Coelen and Wilson impose on their supplyofentrants model a hierarchical structure, wherein the demand for new hires is assumed to be satisfied with reentering teachers, if enough are available, and only then with new entrants if unfilled vacancies still remain. It is not clear what the empirical basis is for this assumed order of precedence nor how the formulation holds up in practice, particularly in years in which relatively few teachers were hired. Moreover, the fact that the number of noncertificated new entrants is projected as a constant percentage of all new hires raises the possibility that such persons could be given priority over persons with certificates, according to the model, in the teacher hiring process.
Finally, a word is needed on the treatment of the supplydemand balance in the Massachusetts analysis. The CoelenWilson model differs from other state models in that it does not yield separate demand and supply estimates but instead offers projections of teacher hiring and employment that are supposed to reflect both demand and supply factors. The employment projections depend, however, on the simplistic assumption that the number of teachers employed in each future year will be the lesser of the projected number demanded or the projected number supplied.^{30} Specifically, if the supply projections, based on the previously described sets of projected entry rates, fall short of the demand projections, the model ''adjusts" by resetting teacherpupil ratios at levels consistent with projected supply. The need for such an adjustment could easily have been avoided simply by assuming, not unrealistically, that the number of noncertificated entrants will expand or contract as needed to balance supply and demand. Alternatively, it could have been assumed that salaries and/or certification standards would adjust as needed to compensate for any supplydemand imbalance. The present assumption, taking the number of teachers in the supply as literally fixed at a level consistent with past entry rates, seems far too artificial and mechanical to be an acceptable representation of how the teacher market operates.
The MISER Multistate Model
According to preliminary specifications for the MISER NEDSAD model (Coelen and Wilson, 1991b), the supply of entrants will be projected, as in the 1987 Massachusetts analysis, by applying estimated transition rates to numbers of certificated persons in various supply pools. These transition rates will be differentiated primarily according to the durations of such persons in their respective pools—that is, the number of years since certification, in the case of persons who have never taught, and the number of years since leaving teaching, in the case of persons who have previously taught in the state in question. The specifications do not allow for entry rates to vary in relation to such other characteristics of potential teachers as age, sex, and subject specialty. Coelen and Wilson say that they will allow for effects of relative salaries on rates of entry and reentry into teaching (this is the only explicit behavioral element of the model), but their proposed treatment of the salary variable is both unusual and restrictive. They postulate that relative salaries affect entry rates only in periods of "supply shortage" and, moreover, that such periods can be identified from data on the percentage of entrants who have no teaching experience (the premise seems to be that inexperienced teachers are hired only as a last resort). They propose, therefore, to construct a model in which entry rates are determined by relative salaries in socalled shortage periods and by total hiring rates in socalled demanddominated periods. This approach, they believe, will allow them to distinguish between supplyside and demandside influences on hiring of teachers. They also indicate that the model will deal with interstate flows of teachers, but the details are not spelled out. Finally, although the NEDSAD data base consists of multiyear files on individual teachers, Coelen and Wilson intend not to use these individuallevel data sets to analyze entry patterns but rather to limit themselves to estimating entryduration relationships from aggregated data on average entry rates of persons in the various duration categories.
These model specifications are problematic in multiple respects, but it is not feasible at this preimplementation stage (nor would it be fair) to offer a detailed critique. I confine myself to noting some major concerns. First, at the conceptual level, the MISER framework does not distinguish clearly between influences on teacher supply and influences on entry into teaching. Although Coelen and Wilson recognize that the effects of demographic factors, salary levels, and other influences on supply behavior may or may not be reflected in entry rates, depending on what is happening on the demand side of the market, this awareness is not reflected in the proposed modeling technique. Second, the suggested method of sorting out demandside and supplyside effects on hiring is difficult to reconcile with labormarket economics. The eitheror, surplusorshortage characterization of the teacher
market is untenable; shortage cannot be defined meaningfully without reference to teacher quality; and the mix of entrants per se bears no necessary relationship to the tightness of supply (for instance, even with no experienced teachers to hire, there could be more than enough newly trained teachers to staff the schools). Third, it is especially troubling that Coelen and Wilson seem to have decided in advance not to include potentially important influences on entry rates in their model. Specifically, the decision to model entry rates solely as functions of duration in the supply pool is hard to defend in the face of evidence that such rates also vary by age, sex, and subject specialty (see "research on the supply of entrants," below). Finally, MISER's decision to construct only relatively crude aggregative models of hiring seems inexplicable, considering that largescale individuallevel data bases are available and that Murnane, Grissmer, and others have convincingly demonstrated both the feasibility and the value of modeling supply behavior at the individual teacher level.
SurveyBased Supply Projections: The Connecticut Approach
The Connecticut teacher supplydemand study (Connecticut Board of Education, 1988) provides an unusually rich body of descriptive information on sources and characteristics of entering teachers but offers only a simple and incomplete set of projections of future supply. What makes these projections noteworthy, however, is the method used to determine how many persons in a particular category should be considered "in the supply." Instead of calculating historical entry rates, the Connecticut analysts chose to base their estimates of each group's participation rate on survey findings concerning potential teachers' jobseeking behavior and intentions. The Connecticut method, therefore, constitutes a third approach to projections, distinct from the two methods described previously.
The Connecticut definition of supply is linked to certification but, unlike the Nebraska and South Carolina models described earlier, the Connecticut model does not equate numbers of certificate holders with numbers in the supply; instead, it focuses on the propensities of different categories of certificated persons to seek teaching positions. The analysis focuses primarily on two broad categories of prospective teachers: new certificants and members of the reserve pool. The latter is defined simply as the stock of persons certificated in earlier years but not currently teaching. It includes both former Connecticut teachers (including some on leave from school districts) and persons previously certificated who have never taught. The newcertificant category includes potential teachers of four types: new Connecticut college graduates who have just obtained certification, earlier Connecticut graduates now applying for certification, individuals newly trained in other states and immigrating to Connecticut, and experienced immigrat
ing teachers. (In recent years, only about 30 percent of entrants have come from the new certificant category; the remainder are from the reserve pool.)
In separate surveys of samples of new certificants and former Connecticut teachers, respondents were asked about their actual job market behavior and the results (whether they had applied for Connecticut teaching positions, whether they had been accepted, and why, if accepted, they had not taken jobs) as well as about the likelihood of their applying for teaching jobs in the forthcoming year and within the next five years. The details of how the survey results are used to project the future supply of entrants are hazy, but what is clear is that the reported rates of applying for jobs are the basis for shortrange (one year) projections of likely numbers of applicants from the two groups covered by the survey.^{31} It is not clear to what degree these rates are differentiated by subject specialty. The sample sizes, except in the large area of elementary education, appear too small to support subjectspecific estimates. The rates at which respondents expressed interest in applying in the future are reported, but it is not indicated that these rates were used to make longerrange projections. No complete projection model is presented. For instance, no method is described for estimating the future size of the new certificant pool, even though a relatively simple trend analysis would probably suffice for that purpose. Also, no projections are offered for members of the certificated reserve pool who are not in the formerteacher category. Given all these gaps, the Connecticut analysis cannot be offered as an example of a complete projection model. Nevertheless, the notion of using survey data to estimate participation rates merits further attention and development, especially considering the present lack of viable alternatives.
Assessment of Projection Methods
All the models described above clearly have major conceptual and technical shortcomings. None qualifies, in its present form, as an acceptable method of estimating the future supply of new entrants. The question is what, if anything, might be made of each approach in the future. I comment here on the problems with each type of model and the possibilities for further development.
Attempting to measure or project supply solely in terms of numbers of persons with teaching certificates is a conceptual dead end. The problem is that certification is only weakly and indirectly related to teacher supply. The connection between the two breaks down in two respects: first, many people with teaching certificates are not in the supply—that is, they are not interested in applying for or accepting teaching positions. This applies both to newly trained, newly certificated individuals, some of whom demonstrate no interest in actually becoming teachers, and especially to certificate hold
ers in the reserve pool, many of whom could not be induced to enter or reenter teaching under anything resembling current salaries and job conditions in teaching. Second, the supply of teachers is not necessarily limited to certificated persons. In many states, significant percentages of new teachers enter with provisional or emergency credentials or under waivers of certification. Some states have lateral entry, or "alternative certification" provisions, under which people without traditional teacher training can be hired as teachers with little delay. Even where regular certificates are required, it is only in the very short run that the supply is limited to persons who already hold them. The potential supply of certificated teachers two years from now includes not only current certificate holders but also all the people who could obtain certificates within a twoyear period—and who might do so if salaries and other job conditions were sufficiently appealing. Thus certification is, in one sense, too broad a criterion and, in another sense, too narrow a criterion for determining who is in the supply.
Projections of numbers of certificate holders might be useful in conjunction with a model that also projects the rates at which different types of certificated persons apply to teach. Such projections would be needed, for instance, to implement the approach reflected in the Connecticut analysis described above. But comparisons between projected demands for new hires and projected stocks or flows of certificate holders, such as those provided in the aforementioned Nebraska and South Carolina studies, yield very little useful information about the future supplydemand balance.
Analyses and projections of the rates at which various class of potential entrants actually enter teaching are useful for a variety of purposes, but as projections of supply they are gravely flawed. Such projections rest on the unwarranted assumption that the observed rates of entry depend only, or mainly, on the choices of the prospective entrants themselves (supplyside behavior) and not on the hiring decisions of school systems (demandside behavior). But as explained earlier, under a regime of excess supply, such as probably prevails most of the time in all but a few "critical shortage" fields of teaching, the opposite is closer to the truth. Typically, only a fraction of those who seek teaching jobs are hired; the remainder end up in nonteaching activities (or perhaps in teaching jobs in other states). Under normal circumstances, therefore, entry rates per se convey little information about teacher supply. That only 30 percent of the new graduates of a state's teacher training program actually enter teaching, for example, does not signify that only 30 percent were in the supply. A much larger fraction—perhaps 50 percent or even 90 percent—may have been willing to supply their services. There is no way to find out from data on numbers of new hires alone. Similarly, there may be 1,000 members of the reserve pool ready and willing to enter teaching this year, but if school systems want only 200, only 200 will be hired. The true supply is not observable directly
and cannot be estimated from data on the actual number of entrants. In general, therefore, a projection of entrants, either in the aggregate or by category, does not constitute a projection of teacher supply. ^{32} Consequently, it is normally not correct to assume (as is done, e.g., in the previously described Massachusetts analysis) that the supply of entrants in the future will be limited to numbers consistent with entry rates in the past.
The problem of disentangling supply and demand econometrically is very difficult but not necessarily insuperable. In a recent paper prepared for MISER, Ballou and Podgursky (1991) sketch out a preliminary theoretical framework showing how, and under what conditions, a separate teacher supply function might be estimated statistically. Significantly, their work seems to imply that bringing the quality dimension of teacher supply into the model is critical—that is, the possibility of estimating a supply function hinges on being able to observe the qualityrelated characteristics of the teachers hired at different times or in different places. Thus, two key deficiencies of existing models, the lack of a sound method of quantifying teacher supply and neglect of teacher quality, prove to be logically related. It appears that both will have to be dealt with together if either is to be dealt with at all.
What of the surveybased approach represented, in embryonic form, in the Connecticut model? In general, economists tend to resist using such kinds of survey data on the grounds that statements of intent and responses to hypothetical choices are poor substitutes for, or predictors of, actual behavior. In this instance, however, the alternatives are unpromising. The other methods now used in projection models are inadequate for reasons already explained. The standard techniques of econometric supplydemand modeling that one would bring to bear in an analysis of a private labor market seem inapplicable because an essential analytical assumption—supplydemand equilibrium—seems inappropriate to make about the teacher market. Without that assumption, there is no way to estimate supply behavior from information on quantity (number of hires) and price. Using "soft" data, such as survey responses, may be, under the circumstances, a reasonable default option.
One other possibility suggested by the Connecticut survey approach is to base supply projections on more systematically collected data on numbers of applicants for teaching positions. The rates at which members of various pools of eligibles apply to teach is certainly more closely related to supply than are the rates at which they are hired. It is possible, as in the Connecticut approach, to collect application data from individuals, but it may also be feasible, and perhaps preferable, to collect such data from local school systems. Doing so would have the advantage that the data would not be limited to particular predefined categories of applicants—for instance, outofstate as well as instate applicants would be covered. An important
practical problem is that individuals often submit multiple applications, which means that screening for duplication would be required. A more important problem is the conceptual one that application behavior is partly demand determined. Such actions by employers as announcing and advertising vacancies (or simply the fact that hiring is taking place) generate applications, while other behaviors, such as making known that no teachers are needed, discourages them. Therefore, estimates based on application rates are likely to understate supply. Even so, they are likely to come much closer to the truth than would estimates based on the rates at which teachers are actually hired.
Research on the Supply of Entrants
I concluded in a previous section that the most promising work on patterns of teacher attrition and retention has come from academic research rather than from efforts to project the future supply of retained teachers. The same is true, although to a lesser degree, of work on the supply of entering teachers. In fact, the same body of research as is summarized in the attrition section yields, often as a byproduct, some interesting findings about entry behavior and some useful suggestions as to how future research on entry into teaching might proceed.
The component of the supply of entering teachers that is easiest to model is reentry by individuals who have temporarily left teaching jobs. The same longitudinal data on cohorts of entering teachers and the same hazard modeling techniques as are useful for studying attrition can be used to analyze reentry as well. In both their Michigan and North Carolina studies, Murnane et al. (1988, 1989) use these data and methods to relate the probability of reentry to teacher characteristics. With respect to Michigan teachers (entering cohorts of 1972 and 1973), they find that onefourth to onethird of all leavers return. The probability of entry varies with age and gender—young women, in particular, being both the most likely to leave and the most likely to return. Reentry rates also vary significantly among subject fields, with chemistry and physics teachers the least likely to return, elementary teachers the most likely, and teachers of other subjects (including mathematics) falling in between. Similarly, in their analysis of North Carolina teachers (entering cohorts of 1976 and 1978), the same authors find that 30 percent of all leavers during the first five years of teaching eventually return, with women twice as likely to return as men, elementary teachers more likely to return than secondary teachers, and teachers of mathematics less likely to return than teachers of other subjects. In addition, they report that the likelihood of returning is significantly lower for teachers with high NTE scores than for teachers with low NTE scores—one of the few findings yet presented on the quality dimension of the supply of entering teachers.
In the only multivariate study of new entrants into a state's teaching force that has come to my attention, Murnane and Schwinden (1989) use North Carolina data to analyze the rates at which newly certificated individuals enter teaching jobs. This analysis, which covers persons with no prior teaching experience who obtained certificates between 1975 and 1985, uses a logit model to relate the probability of entry to characteristics of the certificants. Among the principal findings are that (1) entry rates were once much higher for women than for men, but the male and female rates have since converged, (2) entry rates vary among subject areas, being highest in special education, lowest in business education, and inbetween in mathematics and science, and (3) entry rates in certain subject areas, including mathematics, chemistry, physics, and English, tend to be lower (among whites only) for persons with high NTE scores than for persons with low NTE scores. ^{33} Also, Murnane and Schwinden cite evidence, albeit less rigorous, showing that both numbers of certificants and certificants' choices of subject specialty appear to be sensitive to demand.
An important implication of these findings for projections of rates of entry into teaching is that both the rates at which newly trained teachers enter and the rates at which former teachers reenter need to be differentiated by subject specialty and in relation to certain teacher characteristics. Applying the same entry or reentry probabilities to everyone is no more reasonable than applying undifferentiated attrition rates to all categories of teachers. Similarly, it appears (although the evidence thus far is limited) that entry and reentry rates vary over time, just as do attrition rates, and therefore that projections based on fixed rates are likely to be unsound.
It should also be recognized, however, that although multivariate models of entry behavior are preferable in several respects, the estimates of entry and reentry rates from such models are no more indicative of supply behavior than are the average entry and reentry rates used in state projection models. In both cases, rates of entry and reentry into teaching may be determined more by school systems' hiring decisions than by the supply behavior of potential teachers. Murnane and his colleagues are, of course, fully aware of this problem and warn explicitly that their models cannot be interpreted as representing supply behavior. Whether anything more can be done in the multivariate framework to reduce the confounding of demand and supply factors is not yet clear. One possibility is to estimate models that contain explicit demand factors (e.g., measures of enrollment or expenditure growth or even aggregate measures of hiring), with a view to controlling for demandside influences on entry rates. To implement such models properly is likely to require a particularly dataintensive form of modeling, in which entry behavior is studied with multiyear files of both employment and certification data.
Prospects for National Projections of the Supply of Entrants
There are currently no national models for projecting the supply of entering teachers. Formerly, NCES offered projections of one element of that supply, the flow of new graduates from teacher training programs, but these have been abandoned as unreliable and, in any event, did not yield estimates of how many new graduates (much less any other potential teachers) would actually enter teaching jobs. Today, however, two other types of national data are available that have some potential to provide information about the future supply of new teachers: (1) the Schools and Staffing Survey (SASS), discussed in connection with teacher attrition, and (2) the National Longitudinal Study (NLS) of the high school class of 1972 and, eventually, other longitudinal surveys of later cohorts. I comment here on the earlier NCES projections and on potential applications of these other data sets.
The Former NCES Supply Projections
The classroom teacher sections of NCES's annual Projections of Education Statistics included, as of 1985, not only projections of teacher demand and teacher attrition but also projections of one element of teacher supply, the annual production of new graduates of teacher training programs. These projections were linked to the NCES projections of numbers of recipients of bachelor's degrees. Specifically, this was accomplished by extrapolating the ratio of newteacher graduates to all new bachelor's degrees according to a loglinear trend equation. The resulting estimates of numbers of newly trained teachers were referred to, in years prior to 1985, as supply projections and compared with projections of the demand for new entrants. If the projected demand for new hires exceeded the projected number of new graduates, the difference was referred to as an anticipated shortage.
NCES eventually ceased comparing projections of new teacher graduates against projections of demand after recognizing that the number of new teacher graduates (1) overstated the supply of newly trained teachers because not all graduates actually sought to become teachers and (2) neglected entirely all other sources of teacher supply—i.e., all reentering teachers and other reserve pool members. It then dropped the projections of new teacher graduates entirely, possibly because of findings that such projections were proving highly inaccurate (Cavin, 1986). Currently, the agency offers no projections of any component of the supply of entering teachers.
Data from the NCES Schools and Staffing Survey
The NCES Schools and Staffing Survey provides considerably richer national data than have hitherto been available on the sources from which
entering teachers have been obtained. Using these data, it should be possible to carry out a detailed descriptive analysis of sources of supply, showing, among other things, how the mix of entrants by source varies among types of places and among subject fields. Such information, though not directly indicative of future supply prospects, is of considerable value in its own right.
Two features of the SASS data base limit its usefulness for developing projections of the supply of entering teachers. One limitation is that SASS is a crosssectional data base, now applicable to only the single year 1987–88. It cannot yet be used, therefore, to establish or project trends. As subsequent rounds of data collection are conducted (the round now under way covers 1990–91), SASS will begin to provide some timetrend data. The second limitation is that SASS provides data on actual teachers but not on nonteaching members of the groups from which teachers are drawn. Thus, the SASS data base, by itself, cannot be used to estimate the rates as which members of particular groups enter teaching or to relate entry rates to particular characteristics of individuals. Combined with information from other sources, however, the SASS data could shed some light on these matters. For example, the SASS data on the educational backgrounds of new entrants (degrees and majors), combined with external data on degrees granted, could be used to estimate entry rates for recent college graduates, broken down by race, sex, and perhaps other characteristics. These rates could then be used to project numbers of entering college graduates in the future. Like other such projections, however, these would confound the effects of supply and demand factors on numbers of entrants and would not constitute true projections of supply.
The usefulness of SASS for analyzing teacher supply from the reserve pool is particularly limited because neither SASS itself nor any other national data base provides data on the size or composition of the pool. However, one aspect of supply from the reserve pool, namely the rate at which teachers who leave teaching subsequently return, may eventually become analyzable. It is now planned, I understand, that teachers in the 1990–91 SASS sample who leave their jobs will be followed up several times, rather than just once as in 1987–88. The multiple followups will make it possible to identify returners and to establish the rates at which leavers resume teaching. Other flows from the reserve pool will be observable with SASS data, but it will not be possible to translate them into rates or to project them for lack of data on the size of the pool itself.
Data from Longitudinal Surveys
Data from NCES's National Longitudinal Study of the high school class of 1972 (NLS72) have already been used to study certain aspects of teacher
supply, and other longitudinal data sets, including High School and Beyond, should also be useful for that purpose in the future. An analysis by Manski (1985), using only the NLS72 data accumulated through 1979, examines, among other things, factors associated with the choice of teaching as an occupation. A more recent analysis by Heyns (1988), using data from the 1986 NLS72 followup survey, examines the rates of entry into teaching of individuals who had completed teacher training as well as subsequent rates of attrition and return to the profession. These analyses are preliminary, aggregative, and descriptive, but the same data base has the potential to support more complex multivariate analyses of factors associated with individuals' entry and reentry decisions. Moreover, when similar analyses are undertaken with the High School and Beyond data (high school classes of 1980 and 1982), it will become possible to say something about changes in entry and reentry patterns over time. Thus, considerable potential resides in the longitudinal data sets to gain new insights into national teacher supply.
OVERVIEW: THE STATE OF THE ART AND PROSPECTS FOR IMPROVEMENT
The State of the Art in General
The main attributes, capabilities, and limitations of current models for projecting teacher supply, demand, and quality can be summarized, in capsule form, as follows:

Teacher supplydemand projection models remain mechanical rather than behavioral, which means that they can project future teacher employment and future numbers of new hires only under the implicit assumption that all influences on demand and attrition (or on trends therein) will remain unchanged.

Because the models do not link estimates of teacher supply and demand to budgets, salary levels, and other underlying causal variables, they cannot be used to predict how teacher markets will adjust or to answer whatif questions about the effects of changing conditions and policies.

The better models now yield much more accurate and detailed projections than in the past of the supply of retained teachers, but projecting the supply of potential entrants (other than reentrants) into teaching remains beyond their capabilities. With the supply side incomplete, the models are of limited use for assessing the future supplydemand balance.

It is now common for models to disaggregate projections by level of education and subject specialty, but the current methods of disaggregating demand are too crude and inflexible to capture the effects of changing curricula and patterns of course enrollment.

The models deal only with numbers of teachers and not with teacher quality and thus cannot be used to address concerns about the future quality of the teaching force.
The following paragraphs deal in more detail with the state of the art in projecting the demand for teachers, the supply of retained teachers, and the supply of potential new entrants and with the prospects for improvement in each area.
Projections of Demand
The prevailing method of projecting demand is still simply to multiply projected enrollments by current or extrapolated teacherpupil ratios. This method yields no information on how the teacherpupil ratios themselves, and hence the numbers of teachers demanded, can be expected to change in the future in response to changing economic, demographic, or fiscal conditions. The fact that teacherpupil ratios are merely stipulated or extrapolated rather than linked to underlying causal factors does not necessarily mean that the models will predict future teacher employment inaccurately, but it does render them only minimally useful as policy analysis tools. There seem to have been no recent statelevel efforts to create behavioral demand models, but NCES has moved to develop such a model at the national level by shifting from a mechanical to a regressionbased projection method. Although the present NCES model falls short in many respects, it represents an important step in the right direction and a point of departure for further national and statelevel modeling efforts.
Models offering disaggregated projections of teacher demand by level of education and subject specialty have now become relatively common, but the methods used to disaggregate by subject are extremely crude. The standard approach is simply to assume that the distribution of enrollment among subjects at each level of education will be the same in the future as it is today. None of the models reviewed for this report is designed to project changes in enrollment by subject (except on an ad hoc basis), not even by so simple a method as extrapolating recent courseenrollment trends.
An important implicit assumption built into all the current demand projection models is that the teacher market, both in the aggregate and in each subject specialty, is and has been in a condition of excess supply. This assumption is reasonable in the aggregate and for most subject areas, but it may be wrong for certain critical fields, such as science, mathematics, and special education, that are of particular interest to policy makers. Wherever the excesssupply assumption is violated (i.e., where unmet demand exists), projections based on actual numbers of teachers employed in the past will understate the numbers of teachers likely to be wanted in the future. Al
though some model developers recognize this problem, none has yet offered a solution.
Although the key to making the demand models more useful is to make them behavioral, a number of lesser improvements can be made within the existing mechanical projection framework. These include, for example, developing disaggregated trend projections of enrollment by subject; taking into account that numbers of teachers do not adjust necessarily adjust immediately or in direct proportion to changes in enrollment and that responses to enrollment growth and enrollment decline may be asymmetrical; and perhaps taking into account additional aspects of student body composition, such as changing numbers of handicapped, disadvantaged, and other specialneed enrollees. However, although such additions might improve the accuracy of demand estimates, they would not do much to improve the models' usefulness for policy analysis.
Making demand models behavioral means, concretely, basing demand projections on multivariate econometric models relating numbers of teachers to the fiscal, economic, and demographic factors that ultimately determine how many teachers school systems are able and willing to employ. The pertinent variables include such things as perpupil expenditure, per capita income or fiscal capacity, the pupil/population ratio, the makeup of the pupil population, and salary levels both inside and outside teaching. It appears that such models will have to be constructed at a level of aggregation one rung below the level at which they are to be used; that is, statelevel demand models will have to be constructed with districtlevel data, and a national model with statelevel data. The necessary data appear to be available nationally and for at least some states. Such modeling seems eminently feasible and could proceed if the necessary resources were made available.
Projections of the Supply of Retained Teachers
Attrition/retention modeling has become the bright spot in the supplydemand projection field. The more sophisticated state retention models now project numbers of continuing teachers by applying agespecific attrition rates to age distributions of teachers, differentiating also by level of education and subject specialty. Moreover, recent research on teacher attrition, although undertaken separately from supply projection efforts, has yielded findings that may soon translate into further improvements in models for projecting numbers of retainees. Among these are findings about malefemale differences in attrition patterns, changes in attrition rates over time, and, perhaps most important, effects of salary levels, working conditions, and other economic factors on exit rates from teaching. Further work on these economic factors is required, but if the estimates of effects of
salaries, working conditions, etc., can be refined, taking them into account would transform the present mechanical (albeit sophisticated) models into behavioral models of teacher attrition.
At the national level, the most recently published NCES projections still rely on crudely estimated aggregative attrition rates, but this is about to change because of the availability of more accurate and detailed estimates from the SASS individualteacher surveys. It will soon be possible to project national as well as state teacher retention on the basis of disaggregated agespecific and subjectspecific attrition rates. Research now under way may also lead to the incorporation of certain behavioral elements (e.g., responses to salary changes) into the national models.
In sum, attrition/retention models are already the most advanced elements of teacher supplydemand models, and further progress (and a widening sophistication gap between attrition models and the other components) appears to be imminent.
Projections of the Supply of Entering Teachers
The lack of adequate methods, or even a sound conceptual framework, for projecting the supply of potential entrants into teaching continues to prevent the development of complete supplydemand models that can be used to assess the supplydemand balance. Although several extant models do offer socalled supplyofentrant projections, the projection methods and the premises underlying them are all gravely flawed and the results are invariably projections of something other than supply. It follows, therefore. that little credence should be given to such projections or, especially, to reports of teacher shortage or surplus based on them.
Many current models offer no projections at all of the supply of potential entrants into teaching. Those that do rely on one or the other of two unsatisfactory approaches. The first approach, equating the projected supply to the projected stock of persons certificated to teach in a state, is easily dismissed. Many certificated persons are not in the supply (i.e., willing to teach under current conditions), and many persons in the supply may not (yet) be certificated. The second approach, projecting socalled supply by applying past entry rates to numbers of persons in various supply pools, is invalid because past entry rates are more likely to have been determined by numbers of teachers wanted (demand) than by numbers available (supply). The fundamental problem, with which most model developers have not yet come to grips, is that if past hiring mainly reflects demand, then supply is not directly observable. Consequently, the current teacher supply cannot be measured, and the future supply cannot be projected, from data on employment and hiring. Additional information is needed to estimate how many teachers would be available to take teaching jobs if hiring were not limited by demand.
A possible exception to the above is that it may be reasonable to construe entry rates for one category of entering teachers, those reentering teaching after having left temporarily, as indicators of the magnitude of that element of supply. Some progress has been made recently in modeling reentry behavior, using essentially the same methods as have been used to model teacher attrition. Similar methods may prove acceptable for projecting the supply of reentering teachers. These methods do not apply, however, to the other major categories of potential entrants—newly trained teachers, members of the reserve pool (other than the temporary leavers), and teachers immigrating from other states.
Satisfactory methods of projecting the supplies of these other categories of entrants are unlikely to be developed soon, but several strategies do seem worth pursuing. One is to attempt to estimate supply from information on numbers of applicants for teaching positions. Such information is collectable either from individuals or school districts. Serious problems would arise in interpreting it, but the resulting estimates, even if flawed, might be substantially better than what is now available. A related approach is to derive supply estimates from survey data on the jobseeking behavior and intentions of persons in selected groups. The surveys could be freestanding or linked to such major longitudinal survey efforts as the NCES High School and Beyond. This method would be subject to all the usual problems of using responses to hypothetical questions to infer labor market behavior, but again, the question is whether substantial gains could be made, compared with the present unsatisfactory situation. A third approach is to attempt to distinguish econometrically between the effects on employment and hiring of demand and supply behavior, but as has been shown (Ballou and Podgursky, 1991), this strategy seems to hinge on solving the teacher quality at the same time. This complication places it firmly in the category of longterm (and longshot) possibilities.
REFERENCES
Ballou, Dale, and Michael Podgursky 1991 Estimating Teacher Supply in a DemandConstrained Regime. Draft paper. January. Economics Department, University of Massachusetts.
Barro, Stephen M. 1986 The State of the Art in Projecting Teacher Supply and Demand. Paper prepared for the Panel on Statistics on Supply and Demand for Precollege Science and Mathematics Teachers, Committee on National Statistics, National Research Council. Washington, D.C.: SMB Economic Research, Inc.
Bobbitt, Sharon A., Elizabeth Faupel, and Shelley Burns 1991 Characteristics of Stayers, Movers, and Leavers: Results from the Teacher Followup Survey, 1988–89. June. NCES 91128. Washington, D.C.: National Center for Education Statistics.
Bureau of Labor Statistics 1986 Occupational Projections and Training Data, 1986 Edition. Bulletin 2251. Washington, D.C.: U.S. Department of Labor.
Cagampang, Helen, Walter I. Garms, Todd Greenspan, and James W. Guthrie 1985 Teacher Supply and Demand in California: Is the Reserve Pool a Realistic Source of Supply? September. Berkeley, California: Policy Analysis for California Education (PACE), University of California.
Cavin, Edward S. 1986 A Review of Teacher Supply and Demand Projections by the U.S. Department of Education, Illinois, and New York. Paper prepared for the Panel on Statistics on Supply and Demand for Precollege Science and Mathematics Teachers, Committee on National Statistics, National Academy of Sciences. Princeton, New Jersey: Mathematica Policy Research, Inc.
Coelen, Stephen P., and James M. Wilson III 1987 Report on the Status of Teacher Supply and Demand in Massachusetts, Massachusetts Institute for Social and Economic Research (MISER). June. Amherst, Massachusetts: University of Massachusetts.
1991a Preliminary Specifications of the Supply Side of State Teacher Supply and Demand Models. Memorandum. June 13.
1991b MISER Demographic Teacher Supply and Demand Model. Memorandum.
Connecticut Board of Education 1988 Teacher Supply and Demand in Connecticut: A Second Look. November.
Gerald, Debra E. 1985 Projections of Education Statistics to 1992–93: Methodological Report with Detailed Projection Tables. July. Washington, D.C.: National Center for Education Statistics.
Gerald, Debra E., Paul J. Horn, and William J. Hussar 1988 Projections of Education Statistics to 1997–98. CS 88607. Washington, D.C.: National Center for Education Statistics.
1989a Projections of Education Statistics to 2000. NCES 89648. Washington, D.C.: National Center for Education Statistics.
Gerald, Debra E., Paul J. Horn, Thomas D. Snyder, and William C. Sonnenberg 1989b State Projections to 1993 for Public Elementary and Secondary Enrollment, Graduates, and Teachers. CS 89638. Washington, D.C.: National Center for Education Statistics.
Gerald, Debra E., and William J. Hussar 1990 Projections of Education Statistics to 2001: An Update. NCES 91683. Washington, D.C.: National Center for Education Statistics.
Gilford, Dorothy M., and Ellen Tenenbaum, eds. 1990 Precollege Science and Mathematics Teachers: Monitoring Supply, Demand, and Quality. Panel on Statistics on Supply and Demand for Precollege Science and Mathematics Teachers, Committee on National Statistics, Washington, D.C.: National Academy Press.
Grissmer, David W., and Sheila Nataraj Kirby 1987 Teacher Attrition: The Uphill Climb to Staff the Nation's Schools . R3512CSTP. August. Santa Monica, California: The RAND Corporation.
1991 Patterns of Attrition Among Indiana Teachers, 1965–1987. Santa Monica. California: The RAND Corporation.
Haggstrom, Gus W., Linda DarlingHammond, and David W. Grissmer 1988 Assessing Teacher Supply and Demand. R3633ED/CSTP. May. Santa Monica, California: The RAND Corporation.
Heyns, Barbara 1988 Educational defectors: A first look at teacher attrition in the NLS72. Educational Researcher April:2432.
Lauritzen, Paul, and Stephen Friedman 1990 CSPD Technical Manual: Procedures for Comprehensive Assessment of Educational Personnel Supply/Demand. CSPD [Comprehensive System of Personnel Development] Special Project. Whitewater, Wisconsin: University of WisconsinWhitewater.
1991 Wisconsin Teacher Supply and Demand: An Examination of Data Trends, 1991. Report prepared for the Wisconsin Department of Public Instruction, Wisconsin Teacher Supply and Demand Project.
Manski. Charles F. 1985 Academic Ability, Earnings, and the Decision to Become a Teacher: Evidence from the National Longitudinal Study of the High School Class of 1972. Working Paper No. 1539. January. National Bureau of Economic Research.
Maryland State Department of Education 1988 Teacher Supply and Demand in Maryland, 1988–1991. September.
Metz, A. Stafford, and H. L. Fleischman 1974 Teacher Turnover in Public Schools, Fall 1968 to Fall 1969. DHEW Publication No. (OE) 741115. Washington, D.C.: U.S. Department of Health, Education, and Welfare.
Murnane, Richard J., Edward S. Cavin, and Randall S. Brown 1985 School district responses to enrollment changes: The direction of the change matters! Journal of Education Finance 10(4)(Spring).
Murnane, Richard J., and Randall J. Olsen 1989 The effects of salaries and opportunity costs on duration in teaching: Evidence from Michigan. The Review of Economics and Statistics 71:347352.
1990 The effects of salaries and opportunity costs on length of slay in teaching. Journal of Human Resources 25(Winter): 106124.
Murnane, Richard J., Judith D. Singer, and John B. Willett 1988 The career paths of teachers: Implications for teacher supply and methodological lessons for research. Educational Researcher August–September:22–30.
1989 The influences of salaries and ''opportunity costs'' on teachers' career choices: Evidence from North Carolina. Harvard Educational Review 59(3)(August):32546.
Murnane, Richard J., and Michael Schwinden 1989 Race, gender, and opportunity: Supply and demand for new teachers in North Carolina, 1975–1985. Educational Evaluation and Policy Analysis 11(2)(Summer):93108.
New York State Education Department 1990 Projections of Public School Classroom Teachers, New York State, 1989–90 to 1993–94. January. Albany, New York.
Ohio Department of Education 1988 Ohio Teacher Supply and Demand. March.
Ostrander, Ray, John Prasch, Robert Stalcup, Cy Yusten, and Linda Zech 1988 Teacher Supply and Demand in Nebraska, 1988–1994. March. Lincoln, Nebraska: Bureau of Education Research and Field Services, Department of Education Administration, Teachers College, University of Nebraska.
Popkin, Joel, and B.K. Atrostic 1986 Evaluation of Models of the Supply and Demand for Teachers. Paper prepared for the Panel on Statistics on Supply and Demand for Precollege Science and
Mathematics Teachers, Committee on National Statistics, National Academy of Sciences. Washington, D.C.: Joel Popkin and Company.
Prowda, Peter M., and David W. Grissmer 1986 Perspective on Teacher Supply and Demand: There is No General Shortage. Paper presented at the annual meeting of the American Educational Research Association. San Francisco, California.
Rollefson, Mary 1990 Teacher Turnover: Patterns of Entry to and Exit from Teaching. Paper presented at the annual meeting of the American Educational Research Association.
South Carolina Department of Education 1990 Teacher Supply and Demand for South Carolina Public Schools, 1989–90 Update. December. Columbia, South Carolina: Management Information Section, Office of Research, South Carolina Department of Education.
Theobald, Neil D. 1990 An examination of the influence of personal, professional, and school district characteristics on public school teacher retention. Economics of Education Review 9(3):241250.
Tuma, John, Antoinette Gifford, Laura Horn, and E. Gareth Hoachlander 1989 Enrollment Trends in Vocational and Academic Education in American Public High Schools, 1969–1987. Berkeley, California: MPR Associates, Inc.
NOTES
Discussion
GUS HAGGSTROM
Stephen Barro has given us a comprehensive, informative, and thoughtprovoking review of conventional wisdom and practice for monitoring teacher supply, demand, and quality. His emphasis is on models, notably supplydemand projection models, that can be used to track the condition of teaching, identify trends, anticipate changes, and provide detailed projections of teacher supply and demand. Adopting an idealistic perspective, he downplays mechanical and demographic models that are "capable only of estimating what will happen in the future if established patterns or trends continue" in favor of behavioral models that can allow for the "effects of hypothetical changes in circumstances on teacher supply and demand." Since examples of the latter do not yet exist, Barro devotes most of his discussion to modeling desiderata laced with provocative critiques of recent efforts to provide the basic information about teachers needed to gauge the breadth and severity of school staffing problems today and in the future.
The italicized terms above are often controversial when used to advocate some methods and denounce others. The term model, coupled with adjectives such as causal, econometric, utilitymaximizing, or structural, can incite reverence or raise a red flag. Noting that model refers to a wide variety of conceptual frameworks in the social sciences, David Freedman, a University of California statistician, called it the mword. Thanks in part to Freedman's efforts to reveal the tenuous underpinnings of some timehonored models in psychology, econometrics, drug toxicity assessment, and census adjustment (e.g., see the summer 1987 issue of the Journal of Educational Statistics and the 31 May 1991 issue of Science), the word model has lost some of its mystique. So has the word modeling, which suggests
modelbuilding. But, to many data analysts, modeling is closer to muddling and refers to the use of largely atheoretical methods, perhaps including modelfitting, for converting raw data into more readily assimilated information in the form of tables, plots, regression equations, and so forth.
Unlike mathematical and statistical models, such as the models for graded social systems discussed in Stochastic Models for Social Processes (Bartholomew, 1973), Barro's review of supplydemand projection models reveals a collage of model fragments, statistical techniques, flow chart imagery, and generalities about relationships. Although Barro cites some studies that make use of mathematical and statistical models, his discussion is not a treatment of models per se, but rather a discourse on modeling considerations.
Despite his theoretical perspective, Barro's discussion of teacher supply and demand is downtoearth and insightful. I especially liked his interesting and revelatory summary of existing studies of teacher demand and turnover. It shows that several states already have data systems that allow continual assessments of their teacher work forces along several dimensions. It also underscores the sorry state of information on teacher stocks and flows at the national level that existed prior to the fielding of the Schools and Staffing Survey. Much of Barro's discussion pertains to narrowly focused assessments, often just tabulations, of teacher attrition based on administrative records at the state level. As he observes, there are numerous definitional problems associated with the terms turnover and mobility depending on the teacher population under study, and there are several kinds of losses and status changes that need to be separated for some purposes. I found his discussions of teacher supply and quality less informative, but those topics present difficult conceptual and measurement issues that make them relatively intractable, except in the abstract.
Although this review of existing work is interesting and informative, the implications for forecasting or monitoring the condition of teaching are not clear. In particular, the paper provides little guidance for creating data bases to support educational planning models at any level of aggregation or for developing data systems on teachers to inform realtime decision making by school administrators.
While few would deny the desirability of having an overarching conceptual framework to guide thinking about the components of a data system and to facilitate bookkeeping associated with codifying data elements, maintaining linkages, and ensuring consistency across subsystems, the need for an allencompassing model is debatable. With respect to Barro's interest in detailed projections of teacher supply and demand, I feel that the task is so highly dependent on the nature of the forecasts and data availability that generalities are academic. Fullblown manpower planning models have their place, but so do simple ad hoc datagathering efforts. To support spring hiring decisions at the district level, for example, estimates of im
pending losses of current teachers derived from reports of principals are bound to be more credible and reliable than projections based on analyses of previous years' losses.
Insofar as longterm projections of teacher supply and demand are concerned, it seems plausible that the yeartoyear flows into and out of narrowly defined teacher categories can be simulated using transition and entry rates estimated from flow data, so that the resulting "model" can be manipulated recursively to generate disaggregated projections several years into the future. However, as demographers have long known, projections of this type are extremely sensitive to shifts in the rates, and the errors build up multiplicatively over time (Keyfitz, 1972).
The main attractions of these models are that they build on irrefutable accounting equations and seem to afford unlimited possibilities for refinements. One way of expressing the accounting equations is in terms of the matrix identity
S(t) = P(t) S(t1) + N(t)
relating the stocks of teachers in year t, denoted by S(t), to the stocks in year t1 and the numbers of "new" teachers, N(t), in year t. Here, S(t) and N(t) are ndimensional vectors, such that the ith component s(i;t) of S(t) is the number of teachers in the ith teacher category; and P(t) is an nbyn matrix of flow rates between categories, i.e., p(i,j;t) is the proportion of teachers in the ith category at time t1 moving to the jth category at time t.
Since the components of S(t) and S(t1) can be defined to include any number of teacher categories (e.g., by state, field, and years of service) as well as loss categories for nonreturning teachers, the components can be extended to fit any data system, real or imaginary. Alternatively, the equations can serve as overall constraints for sets of underlying relations, such as equations linking stocks of teachers to enrollment levels, or equations relating flow rates into certain loss categories as parametric functions of teacher characteristics, school attributes, and economic factors.
The realities of providing all the numbers and estimates needed to fill in the equations inevitably lead to simplifying assumptions about linkages in the flows over time or across teacher categories. In addition to structural simplifications, statistical pooling and smoothing techniques come into play, e.g., procedures for smoothing subsets of the transition and loss rates by logistic regression or survival analysis methodology. Given the variety of constraints, feedback mechanisms, statistical adjustments, and allowances for exogenous factors that can be imposed, it is impossible to generalize about the appropriateness of the models. But, as is clear from the limited successes of largescale computer models of the economy, adding complexity and detail may not improve predictability or credibility. Complicated models of social processes are difficult to justify or explain, detracting from
their utility for informing the public of the nature and quality of the data system supporting the projections.
It is easy to challenge some of the particulars in the methodology used by the National Center for Education Statistics (NCES) for generating the 10year projections published in the Projections of Education Statistics. But Debra Gerald and her colleagues at NCES do more than just list extrapolations from a set of prediction equations. I salute NCES's longstanding practice of listing the time series that feed into the projection equations, documenting the data sources, and providing a full accounting of their methodology, often including three sets of projections (low, middle, and high) to reveal their sensitivity to key parameter specifications. Given the methodological warts and data gaps, including the dearth of information on private schools, readers can judge for themselves whether the information justifies saying, for example, that the demand for new teachers will greatly exceed the number of college graduates planning to enter teaching over the next several years.
In any case, national projections are almost devoid of content about local conditions, because the teacher labor market is fragmented into small segments by state, sector, and teaching field. Hefty salary increases in a large school district or the relaxation of state certification requirements can cause big swings in teacher supply and demand within a state by stimulating the mobility of current teachers or opening up new sources of supply. National projections at best whet the appetite for more definitive information about, say, the qualifications, backgrounds, and employment patterns of mathematics teachers in California public schools. Recognizing that many of them launched their teaching careers in private schools, other states, and other fields, and that some of them have taken breaks away from teaching, we see that monitoring the condition of teaching requires tracking alternate routes into teaching and examining the occupational mobility of teachers, not only across states, sectors, and fields, but also into and out of nonteaching activities. As Emily Feistritzer reports in her Profile of Teachers in the U.S.—1990, more than a third of current public school teachers have had at least one break from teaching, and nearly half of the "new" teachers hired since 1985 have had at least one break.
A number of other demographic and economic factors affect teacher flows at the district and state levels. On the demand side, the states differ markedly in their schoolage population growth rates and their means for funding public education. In California, as a result of a severe budgetary crunch at the state level, some districts have laid off hundreds of teachers and resorted to substantial increases in class sizes. Uncertainties about the possible effects of "school choice" and voucher programs add another layer of unknowns. On the supply side, teacher preparation and certification policies seem to be in a continual state of flux. And little information exists
about the dependence of teacher supply on economic conditions affecting other collegetrained workers in the same area.
It was considerations like these that guided our NCESsponsored work at RAND in redesigning the Schools and Staffing Survey to enhance their utility for making assessments of the condition of teaching across the nation. Our 1988 RAND report Assessing Teacher Supply and Demand, coauthored with Linda DarlingHammond and David Grissmer, spells out our rationale for the linked surveys of districts, schools, principals, teachers, and former teachers that were subsequently implemented. Since NCES adopted the survey instruments that we devised almost without change, most of our specifications of data desiderata for a national data base on teachers can be inferred from an examination of the questionnaires themselves.
The district and private school questionnaires asked for detailed staffing breakdowns to permit estimating teacher stocks by state, sector, level, and field. The school surveys also elicited information on teacher turnover by field that can be aggregated to provide key turnover estimates at any level of aggregation or used to augment the individual data on samples of teachers in the same schools.
In addition to providing individual data for profiling teachers along numerous dimensions, the teacher surveys contained a sequence of items pertaining to the teachers' work histories to identify sources of entry into teaching and to pinpoint the nature and timing of subsequent transitions. Since the items include current and previous year's statuses (fulltime, parttime, itinerant teacher, substitute, etc.), year of entry into fulltime teaching, total years of service, years at present school, number of breaks in service, and main activity during the year prior to current teaching assignment, the items extract the key information needed to support analyses of employment patterns and their linkages to school policies, family characteristics, and economic factors.
As the first step in implementing a oneyear followup survey of teachers and exteachers, we recommended going back to the school representatives in the SASS school sample and asking them to fill out a checklist encoding the current statuses of all teachers who participated in the base year survey, including their reported reasons for leaving and current main activities. In effect, this created polytomous outcome measures for updating the work histories of the base year participants to support detailed analyses of "competing risks" in teacher turnover.
Hence, the SASS surveys were specifically designed to allow analyses of teachers' career patterns along numerous dimensions—attrition, fieldshifting, mobility, longevity, breaks in service, retirement, etc. For the most part, the individual data for these analyses are rightcensored observations of continuing sojourns of teaching activity, i.e., most SASS participants will
continue teaching for one or more years. The data can be analyzed by adapting standard procedures for fitting yeartoyear transition rates (e.g., by logistic regression) or by using nonparametric or semiparametric procedures that have been especially tailored to handle censored failuretime or eventhistory data in demographic and biostatistical contexts (ElandtJohnson and Johnson, 1980; Kalbfleisch and Prentice, 1980; Cox and Oakes, 1984; Tuma and Haanan, 1984).
It is also important to undertake the kinds of analyses of teacher stocks, flows, and utilization needed to assess the condition of teaching across the nation today and to pinpoint shortcomings, trouble spots, and signs of erosion that could worsen over the next few years. Here, I am not just referring to the basic information needed to fill in the blanks in manpower planning models, such as those discussed in Bartholomew and Forbes's Statistical Techniques for Manpower Planning (1979), but also the information about teachers' characteristics, qualifications, working conditions, family status, future plans, and attitudes that is needed to profile the teaching work force along many other dimensions that bear directly or indirectly on teacher supply, demand, and quality.
In my view, the SASS survey instruments were well designed to gather the necessary information to meet these objectives. But merely asking the right questions is not sufficient, because the surveys are restricted to probability samples of districts, schools, and teachers. The requisite statistical infrastructure has to be in place to allow extrapolations from SASS samples to the entire population and to permit linkages with other data bases. For the public schools, NCES's Common Core of Data, the repository of annual censuses of districts, is the main linkage between the SASS files and the population; it provides the sampling frame and subpopulation counts needed to support the sampling plan. The SASS files can be viewed as augmentations to the Common Core, which serves as the central data base on public schools. Since the state data systems and studies that Barro cites are also additions to the same data base, the linkages in the combined data base can be exploited provided that the data elements in the various segments are commensurate. Regrettably, there is no similar census or sampling frame for the private schools, so that statistical underpinnings of the SASS private school data are less firm.
By providing detailed information on the employment patterns of teachers and principals, the SASS files also provide data for examining flows into, within, and out of a sizable segment of the collegeeducated work force. Most teachers, especially those with science, mathematics, and business backgrounds, have highly marketable job skills. Analyses of the flows of college graduates into and out of teaching can shed light on the nonteaching opportunities for teachers, circumscribe pools of workers that have some propensity to enter teaching, and identify economic factors related to
shifts into or out of teaching. The major vehicles for tracking the entire labor force are the Current Population Survey (CPS) and the decennial censuses. The Bureau of Labor Statistics (BLS) relies on CPS data to make the projections of demand by occupation that are cited in several publications, including its two series entitled Occupational Outlook Handbook and Outlook 2000. Perhaps because the complicated BLS projections methodology suffers from some of the expositional problems that I mentioned earlier, it is difficult to determine how the projections of teacher supply and demand were generated, and the information in the Handbook about the outlook for teachers is vague. Nevertheless, the CPS and SASS files provide the data and the linkages for extending examinations of teacher mobility into more general studies of employment decision making in the collegetrained labor force, and this opportunity should be exploited.
REFERENCES
Bartholomew, D.J. 1973 Stochastic Models for Social Processes. Second edition. New York: John Wiley and Sons.
Bartholomew, D.J., and A.F. Forbes 1979 Statistical Techniques for Manpower Planning. New York: John Wiley and Sons.
Bureau of Labor Statistics 1989 Outlook 2000. Bulletin 2352, November. Washington, D.C.: U.S. Department of Labor, Bureau of Labor Statistics.
1990 Occupational Outlook Handbook. Bulletin 2350, April. Washington, D.C.: U.S. Department of Labor, Bureau of Labor Statistics.
Cox, D.R., and D. Oakes 1984 Analysis of Survival Data. London: Chapman and Hall.
ElandtJohnson, R.C., and N.L. Johnson 1980 Survival Models and Data Analysis. New York: John Wiley and Sons.
Feistritzer, Emily 1990 Profile of Teachers in the U.S.—1990. Washington, D.C.: National Center for Education Information.
Freedman, David A. 1987 As others see us: A case study in path analysis. Journal of Educational Statistics 12(2)(Summer):101–128.
1991 Adjusting the 1990 census. Science 252(May):12331236.
Haggstrom, Gus W., Linda DarlingHammond, and David W. Grissmer 1988 Assessing Teacher Supply and Demand. May. R3633ED/CSTP. Santa Monica, California: The RAND Corporation.
Kalbfleisch, J.D.. and R.L. Prentice 1980 The Statistical Analysis of Failure Time Data. New York: John Wiley and Sons.
Keyfitz, Nathan 1972 On future population. Journal of the American Statistical Association 67(338)(June):347363.
Tuma, Nancy B., and Michael Hannan 1984 Social Dynamics: Models and Methods. New York:Academic Press.
Discussion
RONALD E. KUTSCHER
The rosecolored glasses that my remarks will be filtered through are those of an economist specializing in the labor market and, in fact, most of my professional career has been devoted to projecting the labor market, so, what I say has been through that filter.
I really have no quarrel with the paper that Steve Barro has prepared. Most of what I hope to bring to you is explicitly or implicitly in his paper. Perhaps I will rephrase them in a somewhat different way, and by doing so provide some additional insights.
I would like to begin by asking a question. Why are supply and demand projections of teachers being developed? I think we must begin with that question because it is a very important issue in determining how much to invest in improving the models used in these projections. That is, if these supply and demand projections of teachers are to be used for information and indicative purposes only, then the cost to users of errors in the projections may be relatively low. If these projections are used to make very concrete decisions such as expanding teacher training programs or institutions, adding to or constructing new buildings, or adding or subtracting staff, then the cost of errors in these projections becomes much higher. If the cost of errors is very large, then we should be willing to spend more time, money, staff, and effort in exploring means of improving the projections.
The principal cost of improving projections is in defining the types of data needed and in collecting underlying data necessary for estimation of new or improved models. The modelbuilding portion of the added cost is a relatively small part of the total cost. Once you determine what data are
needed to develop the models, and you have collected those data, then the model builder's marginal or additional cost is relatively low. In the interim, however, the data development and the data collection phase will have taken considerable time.
If the cost of errors in the use of these projections is relatively low in terms of these decisions, simple models may be providing us with information that is not all that bad. We must judge which of these two costs is closer to reality. If, however, the costs are quite high and we decide it pays to develop more elaborate models, one has to be careful not to fall into a common trap. That trap is that use of a more sophisticated and theoretically elegant model does not ensure that the accuracy of the projections will be significantly improved.
A more sophisticated model may assist you in other ways, however, by explaining why you were wrong when your projections are in error. Secondly, it allows you to be a classical twohanded economist, because you can run what are labeled whatif simulations. For example, you can simulate whatif revenues drop by three percent, how will that affect the demand for teachers? Or, if demand increases for other professionals, what impact will that likely have on the supply of teachers? Answers to such simulations can be very useful to policy makers.
I am assuming the cost of errors is high and that therefore there is a desire to move to develop more detailed and sophisticated models. I say that because, otherwise, why are we here? Consequently, I will make observations that I think are very pertinent to consider in developing a more elaborate model. First, I am extremely skeptical of what I will label as isolated models. These are models that cover only a small segment of a larger entity. I believe there is considerable interaction within an economy. If I were asked to develop a model for Estonia or for Delaware, I believe it could not be done without taking into consideration everything else that is going on around those jurisdictions. Consequently, in developing better models for projecting teacher supply and demand, I urge that this be done in a broader context, which I will elaborate on in a moment. In addition, we must decide whether these models of teacher supply and demand are designed primarily for shortterm or for longterm projections. This is an important distinction because the fundamental nature of how you build these two types of models is different.
How to proceed? I would urge that a teacher supplydemand model begin initially with a broad aggregate type model that describes the entire U.S. economy. From that starting point, the model would be disaggregated toward the subject matter of concern, in this case the teaching profession. To do so it certainly would be necessary to consider the educational industry, government revenues sources and uses, and publicprivate education distinctions—to name just a very few obvious inclusions.
There are two factors are important in any model building: national trends and local trends. In order to try to model any phenomenon, you must attempt to capture both the impact of overall national trends and the impact of local trends on this phenomenon. If you do one and not the other, you will likely increase your projection errors. Local trends can mean geography or they can mean more specific detail about the segment of the economy in which you have the greatest interest.
After this broad beginning, one can begin to disaggregate toward what you need. If ultimately you are concerned with the teaching profession, you start out overall, but then you begin to disaggregate the model by industry in order to capture other employment opportunities for those educated as teachers, by type of school, by geography and other important factors, which could affect teacher supply and demand. Although disaggregating the model by industry will yield more precision in its estimating equations, there is a concern, however, because you very likely will have more error in the underlying disaggregated data. The disaggregation should not proceed so far that the error in the underlying data overwhelms any additional forecasting power that you have gained by the disaggregation.
What other things would I want to see in this model? Well, Steve Barro and Gus Haggstrom both have talked about wages. I think it is very important that wages be in any model that considers supply and demand factors in teaching. According to the Washington Post of March 23, 1991, the average salary of players in the National Basketball Association is $950,000. Does anyone think this conference on teacher supplydemand problems would be held if the average teaching salary was $950,000? If the situation were reversed, maybe the National Research Council would be concerned about the supply and quality of players for the National Basketball Association. So wages are a very important determinant of both the supply and demand of future teachers. Wages are needed not only for teachers but also for other employment opportunities as well.
An additional aspect is the revenues that will be available. You cannot use some overall income variable to project the ability to hire teachers, because income is not necessarily a good proxy for the tax revenues of state and local governments or the share of those revenues that will be devoted to education. Furthermore, it can be important to look at nonwage aspects of working conditions—benefits, for example. We reviewed the supply and demand issues in nursing recently and it turned out that, while wages was one of the factors, working conditions may have been as important a determinant of supply problems in nursing as wages.
One needs to put all of this into a broader economic context. I think it has been noted a number of times that the supply and demand balance for an occupation in a state in which the unemployment rate is 8 percent is much different than it is in a state in which the unemployment rate is 4
percent. When you examine any occupational shortage, you are very unlikely to find shortages if the unemployment rate in the area is 8 or 9 percent. It is only when the unemployment rate drops and other alternative employment opportunities are available that shortages are much more likely to occur. Thus, it is very important to look at the overall economic situation. Of course, I am returning to a point I made earlier about whether the model was to be primarily of a shortterm (1–3 years) or longterm (5 or more years) in orientation.
From my viewpoint, it is most important to look at other employment possibilities. That clearly is so since some persons educated as teachers can decide to teach or they can decide to go into other fields. Consequently, if you do not know what is happening in these other employment opportunity fields, then it is very hard to know what the supply and demand tradeoff is for teachers.
In commenting on Steve Barro's paper, someone noted that they were surprised that the retention rate for mathematicians was different than it was for other scientists. To me, that can be explained by the fact that the alternative employment opportunities for those educated in the other sciences are much greater than are the possibilities for pure mathematicians. Pure mathematicians have a much smaller scope of jobs that are open to them than do natural scientists. So they are more likely to stay on that job.
An additional point that I would think important to consider in model development is to take into account the types of pupils that are to be enrolled. Pupils with handicaps have different teacher requirements, as may pupils in rural areas. Size of the school and size of the community may be variables that are extremely important in terms of demand and even the supply of teachers. All these important factors should be built into a more sophisticated model of teacher supply and demand.
All of my points are made for consideration in improving models for analyzing teacher supply and demand. Even if these recommendations were followed, one could still not develop a model capable of projecting a numerical shortage or surplus of teachers. Techniques are simply not available to accomplish that and are not likely to be so in the foreseeable future.
I would like to close my remarks with one final observation that may only be related indirectly to the question of teacher supply and demand. There is one important trend that is dominant over the last 30 years in the labor market. Almost every professional group has developed a very large paraprofessional or technical group that goes along with the profession. This development of technicians and paraprofessionals exists for the medical, accounting, engineering, and legal professions.
It is interesting and ironic that this trend is not true in the education industry—at least not to any degree approaching that found in the other professional groups. This fact may have an important implication for sup
ply and demand balance in teaching. We are at the point in the medical profession, for example, at which physicians and dentists are among the slowestgrowing component of the health care delivery industry. It is the technician component that is growing very rapidly. A physician's office 40 years ago was staffed by a nurse and a receptionist. Today that same office may still have 1 or 2 physicians, but very likely will have 4 to 10 trained technicians and clerical assistants. By analogy, if the teaching profession had developed like other professions, a teacher would be responsible for 50 or 75 students. However, each teacher would have several paraprofessionals and clerical assistants in the classroom in order to teach such a large group. I think this may be another important dimension of the supply and demand balance that needs detailed examination.
General Discussion
The general discussion addressed the following topics: (a) theory underlying TSDQ projection models; (b) the reciprocal relationship of TSDQ modeling and data bases; (c) constraints on accuracy of TSD projections imposed by the quality of data used in projection equations; (d) equilibration mechanisms in teacher supply and demand; (e) problems in modeling the supply of graduates from teacher preparation programs; (f) the appropriateness of applying concepts from economics to modeling teacher supply and demand; and (g) the utility of TSDQ models and of state and national data bases for informing policy issues in education. The following paragraphs summarize each of these topics.
Since the Barro paper did not consider theory relevant to TSDQ projection models, a suggestion was made that explication of the theoretical underpinnings of these models would be useful for policy makers who are concerned about teacher supply and demand projections. Although Stephen Barro recognized that such theory was not brought out in the paper, he stated that there is a clear theoretical base for much projection modeling of the teacher force on both the supply side and on the demand side.
On the supply side, theoretical strands from human capital theory, labor market search theory, and labor market uncertainty theory pertain to factors that might influence the decisions of individual teachers to enter or to remain in the profession. On the demand side, economic theory of public fiscal behavior is relevant to decisions of state and local government on spending priorities, including the place of education in such priorities. In turn, the demand for teachers is a function of fiscal capacity and the relative price of teachers compared with alternative expenditures.
The reciprocal relationship of TSDQ models and data bases was noted. On one hand, models are useful in furthering the development of data bases. On the other hand, the development of models is constrained by limited availability of data, especially data of sufficient precision and relevance to determine whether the teacher labor market is supplyor demandconstrained.
The precision of teacher supply and demand projections, even with good models, is limited by the inaccuracy of estimates for key variables used in projection equations, when the variables themselves have to be projected. Since projections of variables such as teacher attrition rates and enrollment growth in public education are imprecise, so will be projections of teacher supply and demand derived from equations that use these variables.
In a world of limited resources, however, projection models will continue to be imperfect. The issue is how to handle this uncertainty. Either sufficient resources will have to be invested to develop projections of adequate precision, or the teacher market will be left to equilibrate on its own in accordance with decisions of individuals in the market based on information available to them at the time.
Equilibration mechanisms have been neglected in TSDQ models. In economics, the longrunning equilibration mechanism is relative wage rates. While this mechanism is working out in the teacher labor market, shorterterm equilibration occurs through the mechanisms of adjustments in teacher quality requirements and in teacherpupil ratios. The definition and measurement of teacher quality continues to be a problem. In teaching, a virtually unexplored, but potential equilibrating mechanism is the hiring of a given number of trained teachers versus the hiring of fewer teachers plus paraprofessional assistants. Good TSDQ models will explicate these equilibrating mechanisms and how they function under conditions of disequilibrium.
Since much of the entering supply of teachers comes directly from output of teacher preparation programs, difficulty in modeling this source was discussed. It was observed that undergraduate students contemplating teaching as a career can delay making a commitment to this profession much longer than students committing to other professions such as medicine and engineering. In essence, it was hypothesized that the pipeline for newly prepared teachers is much shorter than in many other professions and therefore more difficult to model accurately.
Although some doubt was expressed about the validity of this hypothesis, the presumed shortness of the pipeline was viewed as an advantage in that the teacher preparation process, including decisions by individuals to enter it, is very responsive to shifts in demand in the teacher labor market. With increased demand, however, there is the possibility that new supply might quickly overshoot the increase in demand. While the asserted shortness and flexibility of the new teacher pipeline might indeed be valid obser
vations, wide variations in teacher preparation and certification regulations, by state, pose great difficulties in modeling this aspect of teacher supply. This modeling must be done on a statebystate basis.
A question was raised about the utility of applying economic principles of supply and demand to modeling the teacher labor market, since hiring decisions in education typically are made by principals at the school level, and must be responsive to changes in curriculum policy such as increased requirements for mathematics courses. These hiring decisions are strongly affected by school variables such as size. Smaller secondary schools require many teachers who are capable of teaching in more than one field, such as in chemistry and physics, while very large high schools might be able to hire specialized mathematics teachers. Principals consider tradeoffs such as hiring a new teacher in a particular subject matter or reassigning a teacher already on the staff. Similarly, teachers who must instruct in two or more subject areas might be more likely to leave a school for one in which they can concentrate their instruction in their preferred subject. Rather than largescale supply and demand modeling, it might be more productive to study and model teacher flows at the school level.
In the final analysis, the utility of TSDQ projection models depends on the degree to which they inform education policy issues. While there are many problems with models, state and national data independent of models can help clarify a number of policy issues. For example, state and national data can contribute significantly to understanding issues such as the role of minorities in teaching, the quality and preparation of science teachers, and the impact of changes in licensing requirements of teachers. Data can be organized to inform policy debate on such issues, even though such data may not fit well into TSDQ models.