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Z E E B R O A D . A N N A R B O R , Ml 4 8 1 0 6 18 B E D F O R D RO W, L O N D O N W C 1 R 4 E J , E N G L A N D 7917789 S M I T H , NORMAN AN E C O N O M I C INSTRUCTION EDUCATION. MICHIGAN SQHERLED A N A L Y S I S OF T H E I N T HE M I C H I G A N STATE UNIVERSITY, University Microfilms International 300n zeebhoad. ann arbor, mi abiob C O S T S OF S YS T E M OF PH.D ., 1978 HIGHER AN ECONOMIC ANALYSIS OF THE COSTS OF INSTRUCTION IN THE MICHIGAN SYSTEM OF HIGHER EDUCATION By Norman Somerled Smith A DISSERTATION Submi tted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1978 ABSTRACT AN ECONOMIC ANALYSIS OF THE COSTS OF INSTRUCTION IN THE MICHIGAN SYSTEM OF HIGHER EDUCATION By Norman Somerled Smith This thesis examines the instructional cost function in higher education, estimates this cost function for the public institutions in the State of Michigan, tests hypotheses about the nature of this cost function, and discusses the policy implications of the results. Researchers to date have implicitly assumed that the instructional cost function in higher education is linear in all instructional products, that it exhibits no fixed costs, and that the cost of each instructional product is independent of the output of all other instructional products. Each of these assumptions is examined in this thesis. Data provided by the Michigan Council of State College Presidents showing expenditures and educational outputs at the departmental level for thirteen institutions in the State of Michigan over a nine year period were used. These data allowed estimation of total cost functions for six disciplines: Business Administration, Education, Engineering, Humanities, Natural Sciences, and Social Sciences. Norman Somerled Smith The estimation of these cost functions was done using weighted least squares multiple regression techniques. And the form of the cost function estimated included, in each case, terms of the second and third power to estimate non-linearity, interaction terms to estimate dependence among the instructional outputs, institutional dummy variables to estimate institutional differences, and a time trend term to estimate cost increases with time. The conclusions reached through the use of this technique bring into some question the efficacy of the current cost estimation techniques. The assumptions underlying these techniques are found wanting since significant non-linearities are found to exist in the cost function, since significant interactions among instructional outputs exist, and since there is some indication that negative marginal and average variable costs exist within certain ranges. In addition, the conclusions of earlier research are brought into question since there is significant evidence that costs do not increase with student level in all cases. This thesis presents evidence that, in some cases, instructional costs decline as a student moves from undergraduate to graduate work for example. It is concluded that each department is a unique entity that must be closely examined in order to find the true underlying relationships between student level and Norman Somerled Smith costs. And it is also concluded that the use of standard cost accounting techniques have been very misleading in estimating the cost relation, not only for a given department but also for departments in general. ACKNOWLEDGEMENT S I wish to thank the following people: Professor John P. Henderson, Chairman of this thesis committee, for his guidance, help and criticism; Professors Byron Brown, and Aaron Gurwitz for their aid, suggestions and criticism; and especially my wife without whose patience, aid, and encouragement this thesis would never have been completed. TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES V vi CHAPTER I Introduction Why are perceived costs important? Plan of the Thesis Summary 1 4 6 7 CHAPTER II Introduction The Model The Welfare Function The Production Functions The Other Constraints Characteristics of the Model The Hypotheses Assumptions of Present Techniques Effectiveness of Constraints and Decision Makers The Market for Student Labor The Market for Faculty Input The Outputs The Output Measure The Classification ofDepartments Conclusion and Summary 11 13 13 14 16 18 20 20 21 22 25 26 29 31 32 CHAPTER III Estimation of the Instructional Cost Function Linear, Square and Cubic Terms The Dummy Variables The Interaction Terms The Time Trend The Constant Term The Technique The Data 43 44 47 49 49 50 50 51 TABLE OF CONTENTS (Continued) Page The Results Business Administration Education Engineering Humanities Natural Sciences Social Sciences Summary and Comparisons Tests of Hypotheses Present Costing Assumptions Hypothesis One Hypothesis Two Hypothesis Three Model Generated Hypotheses Positive Marginal Costs Marginal Costs Increase with Student Level Summary 51 53 62 72 82 91 100 109 116 117 119 120 123 129 129 133 137 CHAPTER IV Summary and Conclusions iv 140 LIST OF TABLES Table Page 1. Academic Departments by Name and Discipline 2. Cost Function Estimates— Business Administration 54 3. Data Characteristics— Business Administration 57 4. Cost Function Estimates— Education 63 5. Data Characteristics— Education 66 6. Cost Function Estimates— Engineering 73 7. Data Characteristics— Engineering 76 8. Cost Function Estimates— Humanities 83 9. Data Characteristics— Humanities 86 10. Cost Function Estimates— Natural Sciences 92 11. Data Characteristics— Natural Sciences 95 12. Cost Function Estimates— Social Sciences 101 13. Data Characteristics— Social Sciences 104 14. Signs and Significance of the Coefficients 110 15. Average Variable Costs 125 16. Marginal Costs 130 17. Differences Between Marginal Costs 134 v 33 LIST OF FIGURES Page 1. Coefficient Signs and (Dis)Economies of Scale 46 2. Business Administration Lower Division 58 3. Business Administration Upper Division 59 4. Business Administration Masters 60 5. Business Administration Doctoral 61 6. Education Lower Division 67 7. Education Upper Division 68 8. Education Masters 69 9. Education Doctoral 70 10. Engineering Lower Division 77 11. Engineering Upper Division 78 12. Engineering Masters 79 13. Engineering Doctoral 80 14. Humanities Lower Division 87 15. Humanities Upper Division 88 16. Humanities Masters 89 17. Humanities Doctoral 90 CD t—1 • Figure Natural Sciences Lower Division 96 19. Natural Sciences Upper Division 97 20. Natural Sciences Masters 98 21. Natural Sciences Doctoral 99 vi LIST OF FIGURES (Continued) Page 22. Social Sciences Lower Division 105 23. Social Sciences Upper Division 106 24. Social Sciences Masters 107 25. Social Sciences Doctoral 108 vii CHAPTER I Introduction This thesis examines the instructional cost function in higher education, estimates this cost function for the public institutions in the State of Michigan, tests hypotheses about the nature of this cost function, and discusses the policy implications of the results. Researchers to date have implicitly assumed that the instructional cost function in higher education is linear in all instructional products, that it exhibits no fixed costs, that all average variable costs are positive by definition, and that the cost of each instructional product is independent of the output of all other instructional products. This thesis will examine each of these assumptions, all of which are necessary to justify continued use of current cost accounting procedure s .^ The current cost accounting procedure can be summarized as follows: A) Assign the costs of resources used directly in the production of a given instructional product to that product; 3) For each instructional product, sum the costs of inputs used in its production to obtain a total cost for the product; C) Divide the total cost by the quantity of output to obtain an average cost; D) Identify this average cost with the marginal cost for 1 2 that product by assuming that additional units of the 2 product can be produced for the same cost. In this process, most researchers have assigned all instructional costs to one, and only one, of the instructional products: lower division, upper division, masters, or doctoral for purposes of this research. While researchers have noted potential inadequacies in the above procedure, this cost accounting procedure 3 has received the widest use ; and since the undertaking of these cost studies requires significant resources, the results must be valued by the decision makers in 4 spite of potential problems. The empirical results obtained through this cost estimation technique conform well to a priori expectations since they show, almost universally, with student level**: that costs increase doctoral instruction is more expensive than masters instruction, masters instruction is more expensive than upper division instruction, etc. Decision makers have assumed this relationship a priori because undergraduate faculty, those faculty members who would typically be found teaching at institutions with no graduate programs, are usually less expensive than graduate faculty; class sizes are often larger at the undergraduate level; and undergraduate teaching loads are usually heavier than graduate teaching loads. Graduate programs look more expensive than undergraduate programs, priori. and they are shown to be more expensive through 3 the use of a cost accounting technique which uses the same logic and which uses the same assumptions which led to the a priori expectations. It should come as no surprise, therefore, that the empirical results conform to the expectations. Given that many tasks involved in the instructional process can be performed by a well prepared student as well as by a full-time faculty member, it is possible in some instances to substitute working students for faculty members in the instructional process at much lower cost. In some cases, this substitution requires the expertise of a masters candidate or a doctoral candidate although, will suffice. in others, an undergraduate student Since graduate students with the proper qualifications are often not available to a department unless that department itself offers a graduate program, the existence of such a graduate program may lower some costs of instruction by allowing substitution of less expensive graduate assistants for regular faculty members. The savings realized due to this use of students should reduce costs of instruction at the level from which the student worker is drawn and not, as present practice would dic t a t e , at the level at which the student works. The results of multiple regression analysis of the cost relationship presented in Chapter III show that these savings and other factors not taken into account by current cost accounting procedures are indeed 4 significant. Why are perceived costs important? Before the present cost accounting procedures were developed, ja priori expectations indicated that costs of instruction would rise as a student advanced, and these expectations were confirmed by the cost accounting procedures used. Society, the legislator, and the department head assign a value to the various instructional outputs of the system of higher education. Yet, while these will differ and the perception of the costs of producing these outputs will vary depending upon the distance from and knowledge of the production process, a decision is made on the quantity of each output which will be produced. With a given expenditure, the levels of production will not maximize the benefit to society unless the value of the last unit of each output has the same proportionate relation to the marginal cost of each output, or in other words, marginal social benefit (MSB) divided by marginal cost (MC) is equal for all outputs. MSBi 777; — MC . 1 = MSB. V7?" MC . j for all i and j j produced j, i and The estimation ofthe social benefit j both orwelfare function is an internalized estimation and will not change as a result of this research. Researchers and decision 5 makers had and have a priori expectations regarding the cost relationship, and the cost function has been estimated using cost accounting procedures, and the perception of the cost relationship thus derived is subject to change and may well be changed by this present research. If the assumptions under which costs were estimated are not met, the estimation of the relative costs of various outputs may be far off the mark. If these estimates have been faulty and have been accepted as true, it is unlikely that welfare has been maximized. The decision making process may not reach dramatically different conclusions based upon reformulation of the relation between costs and outputs; however, very dramatic changes in policy are possible and may be implied. If economies of scale are sufficiently great, institutions should be consolidated. If resource saving interactions do occur among levels of output, graduate education should be spread more evenly among institutions. Costly antagonistic relationships may exist between and among student level production processes which would result in arguments for centralizing graduate programs more than they are at present. 6 Plan of the thesis In order to examine the cost function in higher education, Chapter II will develop the nature of that cost function from the underlying model which will necessarily include the production functions and relative prices. These will determine the least cost techniques to use in production of a given output— the cost function. The model will include the budget constraint and welfare function as faced and seen by the department head and legislature so that the bases of rational decision making, yielding maximization of welfare, can be developed. the hypotheses to be tested will be developed. Here These include hypotheses regarding economies and diseconomies of scale, interaction among levels, and relationships among marginal costs. Using data provided by the Michigan Council of g State College Presidents , in Chapter III, the author will estimate the instructional cost functions in higher education for six disciplines: Business, Education, Engineering, Humanities, Natural Sciences, and Social Sciences. While this breakdown and the classification of departments into these disciplines can be debated, the groupings used are certainly more homogeneous within disciplines than among disciplines; and the groupings are used approximately as they are here by Michigan State University in forming its colleges, by other institutions, 7 and in various research formats. This chapter presents the tests of the hypotheses developed in Chapter II as well. It is important to note that the model used is simplified and abstract— no account is taken of political questions in the allocation of funds nor is the inter­ action among differing perceptions of the welfare function at the various decision levels taken into account— and there is a lack of certain data which has required the use of qualitative rather than quantitative measures to account for quality, non-budgeted research output, and non-budgeted public service output. These simplifications and abstractions limit the results. Chapter IV presents the research and policy implications of this research. Summary Until 1972, economics had not dealt with the question of the nature of the cost and production functions in higher education. The model suggested in 1972 by Fox 7 allowed for the fact that graduate students are often inputs as well as products of the educational and research process, the fact that different levels of students often take classes together, function of scale. and the liklihood that costs are a Economic theory has much to say about decisions among various outputs and how these decisions should be made in order to maximize welfare given the welfare function, the production functions, the prices of the inputs, and the budget constraint. Economic theory, therefore, has much to say in the case of educational products and the nature of the cost function associated with them. This thesis will examine a segment of what economic analysis can conclude regarding the nature of the cost relationship in higher education for instructional outputs. This thesis is unique because it is the first to use non-linear, weighted, multiple regression techniques to estimate the instructional cost function in higher education and to investigate the question of interaction among levels, economies and diseconomies of scale, and the underlying nature of the cost function. g The next chapter will employ Jan Tinbergen's theory of economic policy in order to develop a model from which the hypotheses to be tested will be developed. 9 Footnotes Chapter I 1. These assumptions have not been stated explicitly in some cases; however, they are implied by the technique used. If interactions among levels were present, costs of resources used could not be assigned to one of the joint outputs simply because the resources appeared to be used in producing that output. If the cost function is not both linear and without a constant, then average costs cannot be assumed to equal marginal costs. 2. John H. Powell, Jr. and Robert D. Lamson, Elements Related to the Determination of Costs and Benefits of Graduate Education. (Washington, D . C . : Council of Graduate Schools in the United States and National Association of College and University Business Officers, 1972) page 115. 3. Ibid. Chapter 3. 4. If such studies were not considered to be of value to decision makers, the resources necessary would not be assigned to their production. 5. Gordon Ziemer, et. al«. Cost Finding Principles and Procedures; Preliminary Field Review Edition, Technical Report Number 26, (Boulder, Colorado: National Center for Higher Education Management Systems at W I C H E , November, 1971) page 214. two references found in this and footnote two The 10 describe presently utilized costing techniques and proposals for improvement as well as providing the history of such endeavors. 6. Unit Cost Study: Instruction and Departmental Research. 1962-6 3 . 1963-6 4 . 1964-6 5 . 1966-6 7 . 1968-6 9 . 1970- 7 1 . (Lansing, Michigan: Michigan Council of State College Presidents, December 19— ). 7. Karl A. Fox, editor, Economic Analysis for Educational Planning. (Baltimore and London: The Johns Hopkins University Press, 1972) Chapters eight and nine. 8. Jan Tinbergen, On the Theory of Economic Po l i c y , (Amsterdam: North-Holland Publishing Company, 1952). CHAPTER II Introduction This chapter develops the theoretical model and from it generates hypotheses to be tested in Chapter III through examination of the instructional cost function in higher education which will be estimated there. The theoretical model is based on the standard economic assumption of the rational, maximizing decision maker. This decision maker will attempt to maximize welfare as he reacts to the resources, constraints, prices, production functions, and the welfare function itself. Decisions affecting higher education within a state system of institutions are made at many levels; and at each of these levels, a different decision maker or set of decision makers is involved. makers, Each of these decision it is assumed, attempts to maximize welfare. At the state level, successful maximization implies that the last dollar spent on each educational institution and the last dollar spent on other state activities will produce equal increments in welfare. At the institutional level, the last dollar allocated to each department will produce equal welfare increments; and at the departmental level, the last dollar spent on each output will produce equal welfare gains. 11 12 Welfare gained, it should be noted, is assumed to be a function of many local factors including individual opportunity costs as well as the nature of the program or output produced. Therefore, the above equalization at the margin does not imply that the marginal cost of teaching students at different institutions need be identical. Differences among students imply differences in opportunity costs for those students, and differences in institutional locations also imply differences in individual and societal opportunity costs. These differences also imply potentially different welfare gains. But it will also be assumed, that within a given department, welfare is not a function of the individuals served. In addition to assuming the welfare maximizing decision maker at the departmental level— departmental data will be used to estimate the instructional cost function since the department is the level at which many of the decisions regarding the distribution of resources among approved outputs are made or implemented and is the level at which results can be most easily seen— all departments in similar disciplines will be assumed to be on the same production function and face the same set of prices for inputs into the production process. This will imply that these departments will have the same cost function for a given discipline. 13 The Model The model developed here will be based upon the theory of economic policy as stated by Jan Tinbergen. Such a model has three parts: 9 1) a welfare function, 2) production functions, and 3) the other constraints within which the decision maker must operate. The Welfare Function Welfare will be assumed to be a function of the outputs of higher education. These outputs are the target variables upon which welfare (W) depends or: W = W (C^, o 2 , where there are m outputs. These outputs (0^) of academic departments in institutions of higher education include research, public service, and instruction, each of which can be broken down into numerous sub-categories. This thesis deals specifically and in a quantitative manner with the instructional outputs broken down into four sub-categories by student level: lower division (freshmen and sophomores), upper division (juniors and seniors), masters, and doctoral instruction. The welfare function itself is assumed to be continuous with continuous first and second partial 14 derivatives such that: b\I/ bo. 1 and <^W2/ d 20 ± — O, ' for all i S O, for all i The Production Functions There are r potential inputs (1^). Since it is the case that certain of the outputs of higher education are also inputs input. 10 let IQ reoresent 0. when O. is used as an Separate terminology is necessary since the units of input and output are likely to be different— output in student credit hours (SCH) and input in student work h ours. Output is a function of the inputs, and the production functions, one for each output, are: where: a) q + m = r, m and b) and where: are the maximum proportions of 15 input to output at each student level i or Max (IQ /0^). The value of limits the amount of input IQ available from a given quantity of output of instruction at level i. Or, in other words, each student will work as an assistant only a limited amount of time per week and the maximum proportion that the market will maintain between working hours and credit hours on the average at a given student level is represented by . is clearly not fixed in value since it is likely to be a function of the quality of student in a program as well as other factors. For example, very high quality students are more likely to qualify for forms of student aid not requiring work and are therefore less likely to wish to work implying a low Q^. With a given student population mix and a given student wage rate, (the wage rate or limits are often imposed on a department) there is a maximum amount of student work obtainable, and where 0. is zero i °i is also zero, Constraint a) above, q + m = r, is simply the statement that there are r outputs of which q are not related to any instructional output and m are related to one of the instructional outputs. 16 The Other Constraints The other constraints under which the decision maker in higher education must operate will be assumed to be three: (A) B * P H4.rr where I. = ■»i (B ) 0k i 1 1 , J*t 3 +<5 » 1 I I. Kal 1 j- k = 1, . . . , m X J-, I i . 1 ....... m i3f Where B is the budget imposed on the department, the prices or wages'^ of the inputs, are is a limit on output k imposed from outside the department, and constraint (C) holds where the department is the only source of 0^. The constraints listed in (B) include constraints on curricula, student body size, student body makeup, availability of specialized physical facilities, authorization to offer programs at given levels, and so on, where is the imposed maximum output. All of these constraints must be imposed on the departmental decision maker• The model can now be summarized as: 1. Aggregate welfare depends upon the quantity of instruction produced. 2. Output is a function of the input levels. 3. Output is constrained by budget, prices of inputs, legal limits on output, other constraints imposed from outside the 17 department by the college, university, or outside agencies, and by the production functions. Since this research deals strictly at the departmental level, and since it is typically the case that departments are more limited in their ability to limit their outputs to particular individuals, the assumption that welfare depends upon individuals served has been dropped at the departmental level as noted above. It is clear that welfare is a function of the individuals served but this modification is necessary to deal with the problem at hand because no data on individuals served are available. If this assumption had not been dropped, the welfare function would have read: n for n individuals and m outputs. If the decision maker does not perceive welfare as a function of the individuals served, then it is legitimate to sum all outputs (0. ) across individuals 1j to obtain homogeneous output of each type with respect to individuals. For purposes of this research, and most often in reality at the departmental level, the decision maker is assumed to be blind to individuals served and 18 the welfare function returns to its original form: w = W(0 l , o 2 . 0m ) Characteristics of the Model The following will be characteristics of the model: A) <^W/ B) c)CK/ <^0^ ^ 0, for all i. for j* The above model implies the following total cost function at the departmental level: TC — TC ^ 2 • * •*» ® m ' ^1' ^2 ' * * * ' where CK is the quantity of output i and or marginal cost of input j. ^r^ is the price It will be assumed, again to simplify the model, that relative prices remained constant during the period of the study. 12 This assumption allows the restatement of the total cost function as follows: TC = TC (0lf 0 2 , 0^; P*) where P* is the price level with relative prices constant. For those programs offered and where no constraints except the production functions are effective, the first 19 order conditions for a welfare maximum at the departmental level a r e : ^W/ ^0. *^Oi/ c>I. £w/ c)0k *^Ok/ d I3 P . PT for all i, k such that j and L are actually used in the production of i and k respectively. The second order conditions for a welfare maximum require alternating signs for the determinants of the principal minors of the relevant bordered Hessian. 14 If no constraints other than the technical constraints are imposed and the decision maker is a successful maximizer, it is necessarily the case that c) TC/ 0± at the welfare maximum. ^ 0 Since it is known that constraints are imposed on departments from outside on outputs and on inputs, these conditions need not hold in fact. If constraints were placed such that they did not interfere with the attainment of welfare maximum, then these conditions would hold. do not constrain, achieved. That is, if constraints then the welfare maximum may be 20 The Hypotheses The cost function developed above will be estimated in Chapter III and the estimated function will be used in order to test the hypotheses developed below. Two sets of hypotheses will be tested: 1) the assumptions implied by present cost accounting techniques and 2) the effectiveness of constraints and departmental decision makers in maximizing welfare. Assumptions of Present Techniques The point was made in the introduction that the following assumptions were implied to this thesis by the cost accounting techniques in common use to estimate instructional costs: 1) There are no interactions among levels of instructional output— all outputs are independent. 2) There are no economies or diseconomies of scale in production. 3) Negative average variable costs are not possible in any range of output by definition. These three assumptions will be tested after the instructional cost function is estimated. 21 Effectiveness of Constraints and Decision Makers Since all cost estimates produced for decision making agencies have shown that costs increase with student level and have shown all costs to be positive, it is implied that, at present output levels, all outputs are valued since they continue to be produced and it is also implied that outputs are valued more highly at the margin as student level increases. If constraints placed upon departments are not effective in preventing the reaching of a welfare maximum and if departmental decision makers do not make errors in the allocation process, then the above model implies that, at output levels encountered, marginal costs of instruction must be positive at all levels of output and that marginal costs must increase with student level. In testing these two hypotheses, the criteria really test the degree to which error has or has not been made— errors at the departmental level in allocating resources or at the higher levels in restricting outputs. By examining the constraints on a given department, it may be possible to identify specific errors in the constraints applied, but this will not be done in this thesis. By looking at constrained outputs and finding that marginal costs are in some instances negative or by finding that marginal costs are not increasing where decision makers believe them to be increasing with student 22 level, the constraints and the decision makers can be found lacking in that welfare has not been maximized— welfare as the decision makers view welfare has not been maximized. The following two hypotheses will then be examined empirically to ascertain the effectiveness of the constraints and the quality of the decision making: 4) Marginal costs are nowhere negative for constrained outputs. 5) Marginal costs increase as student level increases. While there is reason to believe that interactions among levels of instruction are likely (hypothesis 1) and while economies or diseconomies of scale in instruction also seem likely to occur (economies of scale seem particularly likely to occur at the lower levels of production— at the undergraduate level) (hypothesis 2) and while negative average costs are possible when interactions occur (hypothesis 3), the model and welfare maximization imply that all marginal costs will be positive and that marginal costs increase with student level when taken in conjunction with past data and past decisions. The Market for Student Labor A significant portion of the argument above turns on the use of students as inputs into the instructional process; and therefore, it seems proper to examine the 23 nature of the market for student labor. The demand side should be quite straight forward with the quantity demanded a function of price, ability, government subsidies, alternate sources of the same expertise, and so on. The supply side should be relatively easily understood also. The student's utility function is: U = U (I, Y, C, E, 0) where I is a vector of factors associated with instruction and certification offered by a given institution such as courses offered and perceived quality, Y is a vector of income opportunities associated with employment, C is a vector of costs incurred in the educational and work process, E is a vector of extracurricular factors associated with an institution and its location, and 0 is a vector of all other factors. It is clear that a student's willingness to work for his institution, once the institution is chosen, depends upon other opportunities for work available to him. Ceteris paribus. most students will prefer working for their institution for the following reasons: a) travel time is minimized, b) work can more easily be scheduled between classes, c) work can more often be done with friends and peers, d) work provides contact with faculty and administrators and other students and is often school related. 24 The graduate student in particular may prefer working as a graduate assistant in that experience can be gained and developed and a) research and teaching b) research topics can be c) familiarity with the academic environment can be developed. These factors tend to reduce the salary necessary to attract student labor below the salary necessary in the non-university marketplace. Thus it should be no surprise if highly talented students with strong backgrounds in particular fields are willing to work at what appear to be less than equilibrium wages. If institutions recognize the full productivity of graduate students, in particular, the market should produce a situation where marginal product of graduate students divided by their price should equal marginal product divided by price for all other inputs. Competition among institutions for assistants should raise the wage to equilibrium. If certain institutions are prevented from competing for these students; however, even if those institutions in the market pay the equilibrium wage, it would be expected that the prevailing student wage rate would rise if all institutions were allowed to enter the market and compete for graduate students and for graduate assistants. This does not mean that the wage would necessarily rise to match wages paid outside the academic area, but certainly the wage would be expected to rise for those categories of students where competition has been limited. Universities may actually 25 be willing to pay higher wages than external agents in some instances because of lack of mobility of the specialized resources needed. The Market for Faculty Input It is important to realize that student employment is not the only argument for the existence of inter­ relationships between and among instructional levels. Many factors enter into a faculty member's utility function and among these are: the quality of student taught, the opportunity for student assistance of high quality, and prestige as well as the opportunity to interact with high quality faculty members and to utilize sophisticated research techniques. These factors indicate that faculty may prefer to work at an institution with higher level programs in a discipline, all other factors equal. supply side effect is not necessary; however, This since there are other factors which may reduce interest in working at institutions with more advanced programs. On the demand side, institutions with graduate programs may be willing to pay more than institutions without graduate programs for a given faculty member since the probability of research grant receipt is higher at the graduate institution given its typically greater prestige and probable superior research support activities and facilities. The receipt of a research grant can 26 effectively reduce the cost of instruction since the overhead funds may more than cover actual expenses incurred while, at least on occasion, it is also the case that instructional activity is engaged in even when salary is being fully paid from the grant. It may well be that faculty members are, in effect, more productive for institutions with graduate programs than for institutions without programs at a given level. Whether savings are realized by paying lower nominal wages due to more attractive working environment or paying lower real wages (while the faculty member receives higher real wages), there is a possibility that the presence of a program at a given level, more probably at the graduate level than elsewhere, may actually lower costs by affecting the cost of faculty inputs. The Outputs The outputs with which this thesis deals quantitatively are strictly instructional but dummy variables will be used to account in a qualitative sense for other products of academic departments including research and public service. The instructional products, as noted earlier, will be classified as lower division (freshmen and sophomores), upper division (juniors and seniors), masters level work), (this category includes graduate professional course and doctoral. Research and public service were 27 excluded as quantitative variables because of lack of consistent, adequate, or reliable data. This lack of data is clearly a problem; however, it is argued below that it is not crucial for the following reasons: A. At least part of the incentive for undertaking and completing research and public service is future monetary and non-monetary compensation both inside and outside the university. This will have a tendency to lower the current cost to the university of research done by the faculty. The same cannot be said of teaching activity generally. B. In compiling the data used here, The Michigan Council of State College Presidents made every effort to exclude expenditures for activities other than teaching wherever possible. Some research and public service activities are separately budgeted and those expenditures were definitely excluded from the expenditure data used here. It is clear, however, that some research and public service expenditures may be included and should be accounted for. C. It is not clear that instruction, within the instructional budget at least, is always done at the expense of research or public service activities. In order to get a faculty member to reduce research output as a result of increased instructional load (although that is unlikely to be a goal), the marginal utility of research effort must be reduced relative to the marginal utility of instructional activity. Since much of the incentive to perform research comes from outside the university and since instructional effort per student and per course can vary significantly, increased instructional responsibility may not significantly change research output in the short run or in the long run. The additional students may simply have to spread the faculty member's instructional time a little more thinly among themselves. 28 An institution can move from a research orientation to a teaching orientation by changing its faculty members, but it is not clear that even this move will produce more instruction since there is a quality dimension to instruction as well as a strict quantity dimension. Given a particular student body type and the instructional needs implied, there is a significant chance that teaching load may be determined largely on the basis of decisions measuring opportunity cost in terms of instruction and instructional quality rather than in terms of other university outputs. These arguments do not indicate that research is necessarily a free good in terms of instruction as measured for this research, but there is clearly the possibility that research is less expensive in terms of instruction than some may believe. 16 Since the factors which have not been measured which determine costs will be allowed to enter the analysis in terms of a qualitative variable, and since there are no consistent and reliable data available, the problem of dealing with the costs of research and public service activities quantitatively will need to await future research. The dummy variables will represent the difference, on the average, in costs between each institution and Michigan State University, the base institution. This will compensate for the lack of data on other outputs to some degree. The efficiency with which this is done will depend upon the homogeniety of the departments within a given discipline and institution. If there is complete 29 homogeniety with respect to other outputs and costs, then the dummy variable will compensate perfectly for the lack of data. The more heterogeneous the departments in a given institution-discipline, the less well the dummy variable will operate. Given the arguments above which show that the cost of other outputs can be less than might be expected and reasonable homogeneity within the relevant departmental groupings, the use of dummy variables in this case will provide the desired results. The Output Measure An ideal unit for measuring instructional output for use in the production function would include: number of students served, 1) the 2) the gain in educational attainment measured from the beginning of the educational experience to the end, 3) the time retention which would measure the retention of material covered and the expansion of knowledge after the formal educational process for which that process was responsible, 4) a difficulty dimension to measure and compensate for the differential abilities of individuals to learn and for the time spent by the student in mastering the material, and 5) a value dimension to standardize the above measures with respect to the value to the learner or to the society of the material learned. 30 Although it does not meet the above criteria for an ideal measure of educational output, the best available measure of instructional output is the student credit hour which is obtained by multiplying the number of students in a class by the number of credit hours of the class. A class with thirty students which carried four credit hours would produce 120 student credit hours (30 x 4) . This measure does account for the number of students served and it at least partially accounts for the quantity of material covered if it is assumed that more credit hours usually imply more material within a given discipline at a given student level. It does not account for individual differences in student ability nor does it account for time retention. The student credit hour is, however, the currency of academia and therefore does have some consistency if converted to the same measurement system (semester, quarter, etc.). In addition, this is the only measure used in the available data. Other candidates for a measure would include the degree and various measures of attainment gained from a testing program. The degree is a difficult measure since requirements vary widely. And costs are difficult to associate with a given program since many students transfer among institutions and follow widely differing time paths toward the degree. 31 Test results would be appropriate if properly and consistently applied and if the results could be attached to a particular course, or to the university in any way, and therefore to a particular expenditure. All techniques presently available to attach university costs to a given university experience have inherent in them the problems discussed in Chapter I with reference to present cost estimation procedures. In addition, test results of the types needed are not available. While the student credit hour is not ideal as a measure of output, it was the measure accepted by the Michigan Council of State College Presidents as well as many others and is probably the best measure available in most senses. The Classification of Departments into Disciplines This research will produce results for six disciplines: Business Administration, Education, Engineering, Humanities, Natural Sciences, and Social Sciences. These are fairly standard classifications although the criteria for membership are not well defined. In developing the classification of departments by discipline, the following were taken into consideration to some degree: 1) consistency in teaching techniques typically used and in class size, 2) typical collegiate groupings at larger universities, 3) other classification 32 systems which have been developed for use in classifying 17 departments for cost study purposes , and 4) the distribution of the data in the various categories as available from the Michigan Council of College Presidents. Where possible, standard classification systems were adhered to, and when a given department did not fit well into one of the six categories used, in the opinion of the author, the data were not included in the study. All classification decisions were made before regressions were run and none were made after the fact. The entire classification of department titles as they were used in this research is found in table 1. There is certainly room for debate on the classification and some decisions between classifications were not easily made, but on the whole it is felt that the classification system is reasonable. Conclusion and Summary This chapter presents a theory of the way that resources are distributed among the various instructional outputs of a system of higher education as viewed from the departmental level. production functions, Given a welfare function, and the constraints under which the decision makers must operate, this model predicts how much and what types of instructional output will be produced. It could also be extended to other outputs 33 TABLE 1 ACADEMIC DEPARTMENTS BY NAME AND BY DISCIPLINE Name Accounting - Business Administration Aerospace Engineering - Engineering Afro-American Studies - Humanities Agricultural Engineering - Engineering American Thought and Language - Humanities Anthropology - Social Sciences Art - Humanities Art Education - Education Astronomy - Natural Sciences Biochemistry - Natural Sciences Biophysics - Natural Sciences Biology - Natural Sciences Botany and Plant Pathology - Natural Sciences Business - Business Business Education - Education Chemical Engineering - Engineering Chemistry - Natural Sciences Civil and Sanitary Engineering - Engineering Classical Studies - Humanities Computer Science - Engineering Criminal Justice - Social Science Distributive Education - Education Economics - Social Sciences Education Administration - Education Educational Leadership - Education Educational Psychology - Education Educational Sociology - Education Electrical Engineering - Engineering Elementary Education - Education Engineering - Engineering Engineering Mechanics - Engineering Engineering Studies - Engineering Engineering Technology - Engineering English - Humanities Entomology - Natural Sciences Evaluation and Research - Education Family Life Education - Education Far Eastern Languages and Literature - Humanities Foreign Languages - Humanities French - Humanities Geography - Social Sciences Geology - Natural Sciences German - Humanities Great Issues - Humanities Greek and Latin - Humanities Guidance and Counseling - Education Higher Education - Education 34 TABLE 1 (Continued) History - Humanities History and Philosophy - Humanities History of Art - Humanities Hotel Management - Business Humanistic Studies - Humanities Humanities - Humanities Industrial Education - Education Instructional Technology - Education Industrial Engineering - Engineering Insurance - Business Labor and Industrial Relations - Social Sciences Linguistics - Humanities Literature - Humanities Management - Business Marketing - Business Mathematics - Natural Sciences Mechanical Engineering - Engineering Metallurgy - Engineering Meteorology - Natural Sciences Mining Engineering - Engineering Music - Humanities Music Education - Education Natural Science - Natural Sciences Near Eastern Languages - Humanities Nuclear Engineering - Engineering Philosophy - Humanities Physics - Natural Sciences Political Science - Social Sciences Psychology - Social Sciences Reading - Education Religion - Humanities Romance Languages - Humanities Russian - Humanities School of Business Administration - Business School of Fine Arts - Humanities Science Education - Education Secondary Education - Education Secondary English, Speech and Foreign Languages - Education Secondary Mathematics - Education Secondary Social Sciences - Education Slavic Languages - Humanities Social Sciences - Social Sciences Social Work - Social Sciences Sociology - Social Sciences Spanish - Humanities Special Education - Education Statistics - Natural Sciences Teacher Education - Education Theatre - Humanities Urban Planning - Social Sciences Zoology - Natural Sciences 35 TABLE 1 (Continued) Discipline Business Administration Accounting Business Hotel Management Insurance Management Marketing School of Business Administration Education Art Education Business Education Distributive Education Education Administration Educational Leadership Educational Psychology Educational Sociology Elementary Education Evaluation and Research Family Life Education Guidance and Counseling Higher Education Industrial Education Instructional Technology Music Education Reading Science Education Secondary Education Secondary English, Speech and Foreign Languages Secondary Mathematics Secondary Social Sciences Special Education Teacher Education 36 TABLE 1 (Continued) Discipline Engineering Aerospace Engineering Agricultural Engineering Chemical Engineering Civil and Sanitary Engineering Computer Science Electrical Engineering Engineering Engineering Mechanics Engineering Studies Engineering Technology Industrial Engineering Mechanical Engineering Metallurgy Mining Engineering Nuclear Engineering Humanities Afro-American Studies American Thought and Language Art Classical Studies English Far Eastern Languages and Literature Foreign Languages French German Great Issues Greek and Latin History History and Philosophy History of Art Humanistic Studies Humanities Linguistics Literature Music Near Eastern Languages Philosophy Religion Romance Languages Russian School of Fine Arts 37 TABLE 1 (Continued) Discipline Humanities (Continued) Slavic Languages Spanish Theatre Natural Sciences Astronomy Biochemistry Biophysics Biology Botany and Plant Pathology Chemistry Entomology Geology Mathematics Meteorology Natural Sciences Physics Statistics Zoology Social Sciences Anthropology Criminal Justice Economics Geography Labor and Industrial Relations Political Science Psychology Social Sciences Social Work Sociology Urban Planning 38 of higher education without significant modification. In addition to developing the model, the following testable hypotheses have been developed: (1) There are interactions among levels of instructional output with respect to costs, (p. 20) (2) There are economies of scale and/or diseconomies of scale in the production of instruction, (p. 20) (3) There are negative average variable costs within the range of outputs which actually occur in educational institutions, (p. 20) (4) Marginal costs are negative in some instances within the range of actual outputs, (p. 22) (5) Marginal costs often do not increase as student level increases, (p. 22) The above are not the forms in which these hypotheses were earlier introduced, but they are in the form in which they will be tested. These hypotheses will be tested through the estimation of the cost function: TC = TC (01 , 02 , 0 3 , 04 , 0 5 , P*) where 0^ through 0^ are the instructional outputs, Oj. represents all other outputs, and P* is the price level. This total cost function for the departmental level was derived from the model developed in the chapter: 39 w = w (o1# o2, o3, o4 , o5) O, — O, (I, , X , I o •L1 “ B * i . = 1 Zt p . j», O 3+q and i . I = °i J K-' xk o. (Q.) D1 o p .1 . + I«i 1 1 2 Z ) -O Zl *-» °i Jk k = 1, 2, 3, 4, 5 • (O ) * II j>f , i . : i = 1, 2, 3, 4, 5 In Chapter III, the above hypotheses will be tested after the departmental instructional cost function is estimated. 40 Footnotes Chapter II 9. Jan Tinbergen, On the Theory of Economic Polic y . (Amsterdam: North-Holland Publishing Company, 1952) . 10. This may be more accurately expressed as "The existence of certain outputs of higher education imply the existence of certain potential inputs which may not be available unless the first set of generating outputs is produced by a given university." 0, and In will be measured in x U1 different units, educational output at the masters level in student credit hours and masters level graduate assistance in hours worked. 11. Marginal cost should also be included with prices and wages to cover the case where prices are not fixed by an external market. 12. It may also be considered to be simplified in that the list of characteristics purchased when a faculty member is hired is long. Price data would be required for each faculty member or for each faculty type, given an adequate classification system, or for each faculty characteristic purchased. Neither of these is available so that this simplifying assumption is required. 41 13. This is strictly from the departmental point of view. Equalization between departments and between departmental and non-departmental outputs is the responsibility of non-departmental decision makers. 14. James H. Henderson economic Th e o r y : and Richard E. Quandt, MicroA Mathematical Approach. (New York: McGraw Hill, 1958) page 271. 15. Here it is assumed that constraints have been placed under the assumption that the departmental decision makers are not capable of maximizing welfare without such guidance because of lack of information or a differing perception of the welfare function. 16. There is even the possibility that the opportunity to do research may reduce costs of instruction in addition to producing research since performing research is an investment in human capital for the faculty member, research yields some pleasure in and of itself for some, and the research process may provide some non-paid instruction where a faculty member is being paid from a research grant. In other words some faculty members may be willing to work for lower nominal wages— thus saving the institution money— while actually receiving higher real wages with part of the payment in the form of opportunity to perform additional research. possibility seems somewhat remote; however, should be considered. This it 42 17. There are several such systems, none of which met the needs of this research. CHAPTER III In this chapter the instructional cost function as derived from the model in Chapter II will be estimated and the hypotheses developed will be tested. Estimation of the Instructional Cost Function Departmental data obtained from the Michigan Council of State College Presidents X8 covering departments in thirteen institutions in the Michigan state system of higher education over a period of nine years will now be used to estimate the instructional cost function in higher education at the department level. The specific function which will be estimated will be of the form: TC = + Bi+4X i + B i+aX i ) + I B i+12Di i | B i+ 24X lX i + 5 B i+ 27X 2X i + B 30X 3X 4 + + B 31T + B 32 + ^ where X^ is lower division (freshman and sophomore) instructional output measured in semester credit hours, is upper division (junior and senior) output, X^ is masters level (masters and graduate professional) instructional output, X^ is doctoral level instructional output, are twelve institutional dummy variables, T is the time trend variable representing the year number 43 44 from 1 through 9, B ^ is the constant, and £ represents the discrepancy between the planned expenditure-output and the actual expenditure-output relation and the combined effect on expenditures of various noise factors. Linear, Square and Cubic Terms This form of the function containing linear, square, and cubic terms for each variable, allows testing of hypotheses regarding the existence of linear relationships, economies of scale and/or diseconomies of scale. these factors included, Hypothesis With (2) as listed at the end of Chapter II can be tested: There are economies of scale and/or diseconomies of scale in the production of instruction. It is expected a priori that such (d i s )economies may well exist, and if they do exist, that they will disappear as output increases and thus affect only lower output levels. After a certain level, it is expected that the cost-output relation will essentially become linear and marginal (dis)economies of scale initially encountered will disappear. Marginal economies of scale at the departmental level will be expected to disappear at some point after all classes, even the specialized ones, are filled to their maximum efficient size. Marginal diseconomies of scale are not expected to be encountered at all at the undergraduate levels of instruction since it is expected that these, 45 if they exist, will be caused by the higher price implied by faculty who are recruited to teach at the graduate level but also teach at the undergraduate level. small graduate level programs, With such faculty members may be inefficiently used as undergraduate teachers in the sense that they may teach undergraduate courses where a less expensive faculty member would have been hired to teach these courses if the institution did not offer a graduate program. With a small doctoral level program, doctoral level faculty may be inefficiently used in the same sense in teaching masters level courses. In both cases the introduction of a small higher level program may be initially more expensive per unit than it is when it is larger and faculty can specialize at the graduate level where appropriate. This potential inefficiency in the use of faculty members will have its largest net effect in disciplines where there are larger salary discrepancies between faculty that would normally be considered qualified to teach at the graduate level and faculty that would normally be considered qualified to teach only at the undergraduate level, where use of graduate assistants is not sufficient to compensate for this inefficiency, and where more specialties in graduate programs are usually necessary in order to attract students {more specialized faculty members may well imply more faculty teaching at both the graduate and under­ graduate level where this would not be appropriate or 46 necessary if the program were large enough to allow faculty to specialize at given student levels as well as in particular areas of their discipline). It is expected that most undergraduate programs will exhibit non-linear relationships showing marginal economies of scale at low levels of output but will approach a linear cost relationship as output increases. Expectations for graduate programs are not well defined since the cost relationship is expected to be more complicated at the graduate level although any initial (dis)economies will be expected to disappear at the margin. Since it is expected that such (dis)economies of scale will exist and will also tend to disappear, the inclusion of both the square and cubic terms can be used to approximate the true relation as well as providing possible tests for the existence of such (dis)economies of scale at the margin. The table below shows the way in which marginal (d i s )economies of scale will exhibit their existence if the coefficients of the cubic terms are assumed to be small, not insignificant, relative to the coefficients of the square terms. Sign of scjuare term + sign of cubic term + GROWING DISECON­ OMIES OF SCALE — INITIAL ECONOMIES OF SCALE INITIAL DISECON­ GROWING ECONOMIES OMIES OF SCALE OF SCALE Figure 1 Coefficient Signs and (Dis)Economies of Scale - 47 Given the expectations argued for above and Figure l f it will be expected that initial economies of scale will be encountered for undergraduate outputs. If any dis­ economies of scale are encountered, these will be seen at the graduate level although economies of scale are also possible. It will therefore be expected that the signs of the square terms, if significant, will be negative at the undergraduate level and will have indeterminate signs at the graduate level. marginal Since it is expected that all (dis)economies of scale will disappear at larger outputs, it will also be expected that the signs of the cubic terms, if significant, will differ from the signs of the square terms. The Dummy Variables Each institution in the study except Michigan State University is represented by a dummy variable which is intended to account for all institutional differences other than level of output. expected to include: These differences are differences in the accounting procedures and definitions of output levels (these data were gathered so that the differences would be minimized; however, in such complex areas, it is unlikely that such differences can be eliminated entirely), quality of programs, differences in institutional effort in noninstructional areas such as research and public service 48 which have not been separately budgeted, differences in the market prices of productive resources in the widely varying parts of the state in which these institutions are located, differences in student clientele, and all other differences. The use of qualitative variables to account for these differences was required by the lack of reliable and consistent data on the various component differences and the large number of such differences which may occur. It is expected that differences will be encountered among institutions and there is reason to believe that, where significant, these differences are likely to be negative for all institutions except the University of Michigan where the difference is likely to be positive when one is present. This relationship is not implied by the model and is expected to vary from discipline to discipline. These expectations are based largely on expected differences in quality of inputs and differences in effort in non-instructional areas— it is expected that in most instances the University of Michigan will produce the largest proportion of noninstructional output and the other institutions will produce a smaller proportion of non-instructional output than Michigan State University, the base institution. 49 The Interaction Terms As argued above in Chapter II, it is expected that there will be interactions among the student levels of production which will affect costs. These are accounted for in the model by the inclusion of interaction terms in the cost function to be estimated and the significance of these terms will be used to test the hypothesis that such interactions do exist. The sign or significance of these terms cannot be derived from the model since in each case there are potential causes for reduced costs and potential causes for increased costs. are expected to vary among disciplines. terms take the form: Division SCH). These again The interaction B*(Lower Division SCH) • (Upper Inclusion of such terms, and there are six possible combinations, allows the first partial derivative of the cost function with respect to one of the outputs, the marginal cost of the output, to be a function of other output levels. Since this is likely to be the case, there is strong reason for their inclusion a priori. The Time Trend The time trend is present to correct for inflation in the costs of inputs over the nine year period covered by the data. It is expected that the time trend 50 coefficient will be positive and significant for all disciplines. The growth in enrollments during this period was not accompanied by equivalent growth in the number of qualified faculty members which resulted in inflation in the prices of inputs to higher education somewhat in excess of the general inflation rate. The Constant Term The constant term is included in the relation in order to represent any fixed costs, to show the mean effect of any omitted variables, and in order to allow 2 for interpretation of standard statistics such as R . The Technique The above cost function is estimated separately for each of the six disciplines: Business Administration, Education, Engineering, Humanities, Natural Sciences, and Social Sciences, using weighted least squares with the square root of total output as the weight, with total departmental instructional expenditure as the dependent variable measured in dollars, and with outputs as the right hand variables measured in student credit hours. The standard assumptions are made regarding the distribution of error terms after the application of the weighted least squares procedure. 51 The Data The data provided by the Michigan Council of State College Presidents cover departments at thirteen institutions: Central Michigan University, Eastern Michigan University, Ferris State College, Grand Valley State College, Lake Superior State College, Michigan State University, Michigan Technological University, Northern Michigan University, Oakland University, Saginaw Valley State College, The University of Michigan, Wayne State University, and Western Michigan University. They included the years for which the Unit Cost Study was done: 1962-63, 1963-64, 1964-65, 1966-67, 1968-69, and 1970-71. Total departmental expenditures for instruction and the quantity of output is provided in semester student credit hours for each of the four student levels: lower division, upper division, masters, and doctoral. The Results The results of the weighted least squares regressions are found in Tables 2, 4, 6, 8, 10, and 12 for the six disciplines. The dependent variable is in dollars. The first column lists the coefficient estimate, the second column lists the standard error and the third column shows t-values. Two tailed tests are used throughout and significance is defined at the ten percent level 52 although other levels of significance are identified in the tables. Two tailed tests are used even though there are reasons to expect certain signs since in each case the relationships are complicated enough that either sign may result with the exception of the time trend variable where the positive value is definitely predicted but where two tailed tests vs. one-tailed tests does not arise. 19 In addition to t-tests on each coefficient individually, F-tests are presented at the end of each table showing the significance of five groupings of the variables: all linear variables, all square variables, all cubic variables, all interaction terms, and all institutional dummy variables. Data characteristics for each discipline are then presented in Tables 3, 5, 7, 9, 11, and 13. Each of these tables is found directly following the relevant weighted least squares results by discipline. For each variable, the mean and standard deviation are shown. After each table of data characteristics, four figures are found numbered Figure 2 through 5, Figure 6 through 9, Figure 10 through 13, etc. These show the average variable cost and marginal cost functions graphically for each output as they would appear with all variables in and evaluated at mean output for the three other student levels. These exhibit all the possible relationships between these relations and give a better idea of the meaning of the parameter values. 53 Business Administration Departments In Table 2, the following variables are shown to be significant and increasing total costs for Business Administration departments— all measures are in dollars per relevant unit: Linear Upper Division ($42.55), Linear Doctoral ($1136), Doctoral Cubed ($.1842E-02), Time Dimension(25650), Lower Division times Masters ($.9813E-02), Upper Division times Masters ($.1131E-01), and University of Michigan ($.1066E07). The following variables are significant and reduce total cost— again in dollars per relevant unit: Linear Masters (-$94.86), Upper Division Squared (-$.5604E-02), Doctoral Squared (-$2,442), Lower Division times Doctoral (-$.9925E-01), Masters times Doctoral (-$.6834-01), and all institutional dummy variables with the exception of Wayne State University. Each of the five sets of grouped variables is significant using an F-test. And it should be noted that all coefficients of square terms are negative while all cubic terms are positive. Wayne State University is apparently more similar to Michigan State in the Business Administration area than the other institutions in the study and just as expected, those economies of scale found significant (significant negative coefficients on the square variable) are eventually swamped by positive coefficients on the matching cubic variable. In fact, all student levels TABLE 2 COST FUNCTION ESTIMATES— BUSINESS ADMINISTRATION Variable Lower Division (LD) Coefficient Estimate Standard Error 19.84 16.17 t-value 1.227 LD2 -.1715E-02 .2808E-02 -.6107 LD3 .7808E-07 •9268E-0 7 .8425 Upper Division (UD) 42.55* 17.55 2.425* UD2 -.5604E-02* .2792E-02 -2.007* UD3 .1858E-06 .1323E-06 1.404 -94.86** 30.77 -3.082* M2 — .3578E-02 .6218E-02 -.5754 M3 .7850E-0 7 .2303E-06 .3409 Masters (M) Doctoral (D) D2 D3 1136** 300.0 3.789** -2.442** . 7459 -3.274* .1842E-02** .4252E-03 4.438** .3942E LD X UD .6739E-04 .1709E-02 LD X M .9813E-02* .3783E-02 2.594* LD X D -.9925E-01** .30 74E-01 -3.229* UD X M .1131E-01** .40 38E-02 2.801** UD X D .3 796E-01 .3292E-01 1.153 -.6834E-01** .2 343E-01 -2 .916* M x D * ** Significant at ten percent level 1% m 55 TABLE 2 (Continued) Variable Coef f icxent. Estimate 25650.** Time Standard Error t-value 2620. 9.790** Central Michigan University -75200.* 42270. -1.779* Eastern Michigan University -68250.* 32590. -2.094* Ferris State College -106400** 32340. -3.291** Grand Valley State College -189900.** 65060. -2.918** Lake Superior State College -150100.* 79580. - 1 .886 * Michigan Technol­ ogical University -81820.* 40520. -2.019* Northern Michigan University -115700.** 37930. -3.050** Oakland University -86540.* 49450. -1.750* Saginaw Valley State College -179400.* 79900. -2.245* University of Michigan 1066000.** 165500. 6.446** Wayne State University 14190. 33160 .4278 Western Michigan University -55500.* 30780. 1.803* Constant -25560. 41800. -.6115 Significant at the ten percent level 56 TABLE 2 (Continued) Number of observations: 158 Degrees of Freedom: 127 R2 .97511 R2 .96923 F-tests for Groupings of Variables Linear Variables 9.619** Square Variables 4.873** Cubic Variables 5.671** Interactions 4.661** Institutional Variables 2.603** Significant at the ten percent level ** 1% 57 TABLE 3 DATA CHARACTERISTICS— BUSINESS ADMINISTRATION Variable Lower Division (LD) Mean 3288 Standard Deviation 2813 LD2 .1867E+08 .4920E+08 LD3 .1833E+08 .1002E+13 Upper Division (UD) 6526 3282 UD2 .5329E+08 .45 97E+08 UD3 .4930E+12 .5787E+12 Masters (M) 2206 3897 M2 .1996E+08 .6624E+08 M3 .2860E+12 .1210E+13 Doctoral (D) 170 344 D2 146600 388700 D3 .1502E-i-09 .50 78E+09 .235 3E+08 .3061E+08 LD x M •4689E+07 .6843E+07 LD x D 420830 .1045E+07 UD x M .208 3E+08 .4249E+08 UD x D .1751E+07 .3 75 7E+0 7 M x D .1505E+07 •4812E+07 LD X UD $/LDSCH MC AVC 20 10 X Max Min LDSGH 10,000 15,000 20,000 FIGURE 2 BUSINESS ADMINISTRATION LOWER DIVISION 25,000 $/UBSCH AVG X Max Min UDSCH 10,000 FIGURE 3 BUSINESS ADMINISTRATION UPPER DIVISION 15,000 $/MSCH X Max Min MSCH 5,000 10,000 20,000 -15 MC -30 VC -60 -75 FIGURE k BUSINESS ADMINISTRATION MASTERS $/DSCH AVG MC 800 600 200 Min DSGH 800 1200 X Max -200 -Uoo FIGURE 5 BUSINESS ADMINISTRATION DOCTORAL 62 show initial economies of scale which are eventually eliminated, although several of the parameter estimates are not significant at the 10% level. In Figures 2 through 5, it can be seen that Business Administration instruction has a cost function estimate that exhibits textbook characteristic U-shaped cost curves at all output levels but that for the Masters level the entire range of values lies below the horizontal axis. The Doctoral level also lies below the x axis for a portion of the Marginal Cost function. It appears that adding a Masters program or expanding it in the Business disciplines may not involve expenses above and beyond those encountered at the undergraduate level and in fact there is some indication here that a Masters program may actually reduce total expenditures on instruction in the Business area. Education Table 4 presents a somewhat different picture in-sofar as the only institutional dummy variable shown to be significant for Education departments is the University of Michigan which is positive as was expected in most cases. Apparently Education departments are more homogeneous among institutions than the Business Administration departments. 63 TABLJi 4 COST FUNCTION ESTIMATES--EDUCATION Variable Coefficient Estimate Standard Error t-value 58.37** 19.56 2.984** — .1224E-01* .4708E-02 -2.600* .2495E-06 3.337** 14.81** 5.26 2.819** UD2 .3350E-03 .3194E-0 3 1.049 UD3 -.9049E-08* .5 374E-08 -1.684* '37.83** 11.54 3.280** Lower Division (LD) LD2 LD3 .8327E-06** Upper Division (UD) Masters (M) M2 - .6131E-02** .1699E-02 -3.609** M3 .1161E-06* .5138E-0 7 2.260* Doctoral (D) 54.84 .4542E-01** D2 58.36 .9397 .1129E-01 4.025** D3 - .8325E-06 .7821E-06 -1.065 LD x UD -.2406E-02* •1132E-02 -2.126* LD X M .5324E-02* .2659E-02 2.002* LD x D - .1725E-01** .5182E-02 -3.328** UD x M .2977E-02** .4580E-03 6.502** UD x D - .6482E-02** .2043E-02 -3.172** .1731E-02 .1818E-02 .9523 M x D * Significant at the ten percent level ** 1% 64 TABLE 4 (Continued) Coefficient Estimate Standard Error t-value Time 17960.** 2668. 6.734** Central Michigan University 76160. 331700. .1995 Eastern Michigan University 156300. 378100. .4134 Ferris State College 225600. 383800. .5878 Grand Valley State College 174300. 387300. .4502 Lake Superior State College 81520. 418100. .1950 Michigan Techno­ logical University 136000. 414400. .3281 Northern Michigan University 161700. 382600. .4227 Oakland University 171300. 383900. .4461 Saginaw Valley State College 119500. 390400. .3062 University of Michigan 488100.* 283200. 1.723* Wayne State University 220600. 378000. .5837 Western Michigan University 237200 382100 .6207 Constant -247500. 380800. Variable * Significant at the ten percent level ** 1% -.6500 65 TABLE 4 (Continued) Number of Observations: 217 Degrees 186 of Freedom: R2 .99253 R2 .99132 F-Tests for Groupings of Variables Linear Variables 13.1546** Square Variables 439.5261** Cubic Variables 218.6409** Interactions Institutional Variables 3.3865** 11.6783** * Significant at the ten percent level ** 1% 66 TABLE 5 DATA CHARACTERISTICS— EDUCATION Variable Lower Division (LD) Mean 2603 Standard. Deviation 3309 LD2 .1768E+08 .35 71E+08 LD3 .1569E+12 .4149E+12 Upper Division (UD) 13530 16540 UD2 .4552E+09 .8646E+09 UD3 ,1966E+14 .4805E+14 Masters (M) 5820 7192 M2 ,85 36E+08 .1685E+09 M3 ,1642E+13 .4176E+13 Doctoral (D) 1302 3026 D2 ,1082E+08 .3145E+08 D3 ,1047E+12 .36 79E+12 LD x UD ,8192E+08 .16 79E+09 LD x M ,3377E+08 .6608E+08 LD x D ,8306E+0 7 .2543E+08 UD x M ,1342E+09 .3605E+09 UD x D ,5180E+08 .1513E+09 M ,2318E+08 .6 796E+08 X D $/LDSCH 20 MC AVC 10 10,000 15,000 LDSCII X Max Min -10 -20 FIGURE 6 EDUCATION LOWER DIVISION $/UDSCli 20 AVC Min X Max 10,000 20,000 30,000 i|0,V00 MC FIGURE 7 EDUCATION UPPER DIVISION 50,000 60,000 UDSCH 70,000 $/MSCH 100 MO AVC 20 Min X Max 30,000 -20 - 1*0 FIGURE 8 EDUCATION MASTERS $/D3CH 800 600 AVG MC 1+00 200 X Mir/ X Max 0 ■DSCH 10,000 20,000 -200 FIGURE 9 EDUCATION DOCTORAL 30,000 71 The following variables are significant and increase total cost in dollars per relevant unit: Linear Lower Division ($58.37), Linear Upper Division ($14.81), Linear Masters ($37.83), Doctoral Squared ($.4542E-01), Lower Division Cubed ($.8327E-06), Masters Cubed ($.1161E-06), Lower Division times Masters ($.5324E-02), Upper Division times Masters ($.2977E-02), Time ($ .1796E-05), and the University of Michigan ($.4881E06). Significant reductions in cost are produced by the following variables in dollars per unit: Lower Division Squared (-$.1224E-01), Masters Squared (-$.6131E-02), Upper Division Cubed (-$.9049E-08), Lower Division times Upper Division (-$.2406E-02), Lower Division times Doctoral and Upper Division times Doctoral (-$.1725E-01), (-$.6482E-02). Each of the five groups of variables have significant F-test statistics which is of special importance for the institutional dummy variables since only one of the twelve is shown significant using individual t-tests. Again each of the cubic terms has a sign that differs from the corresponding square term; however, in the case of Education, the signs alternate among the levels with Lower Division and Masters showing initial economies of scale and Upper Division and Doctoral showing initial diseconomies. This can be seen by examing Figures 6 through 9 where Lower Division and Masters show U-shaped cost curves and Upper Division and Doctoral show less traditional hump shapes. For Education, three of the 72 student levels show portions of the cost curves falling below the horizontal axis for negative cost effects: Lower Division, Masters and Doctoral. The Lower Division result comes as a surprise as this was not expected a^ p ri o r i ; however, it is possible that those institutions that offer a larger portion of their work at the Lower Division in larger classes and perhaps without as much variety in a more standardized core curriculum approach may reduce their costs by reducing the amount of work done at the more expensive Upper Division. While Lower Division cost curves are higher than the corresponding Upper Division curves at low levels of output, they soon fall below their Upper Division counterparts. It is not clear what the cause of this effect may be, but it may well deserve further examination at another time. Engineering Table 6 lists the results for the Engineering departments and shows that just as in the Education discipline all institutional dummy variables fail to attain significance with the exception of the University of Michigan although as a group they do attain significance. Here the Lower Division Linear term ($57.21), Upper Division Linear term ($39.78), Lower Division Cubed ($.1433E-05), the time trend ($32510) and the University of Michigan ($.1283E06) are cost increasing significant 73 TABLE 6 COST FUNCTION ESTIMATES- -ENGINEERING Variable Lower Division (LD) Coefficient Estimate Standard Error t-value 57.21* 33.93 LD2 — .232 8E-01* * .1173E-01 -1.984** LD3 .1483E-05* .8215E-06 1.806* 39.78* 23.15 1.718* Upper Division (UD) 1.686* UD2 -.3054E-02 .4996E-02 -.6113 UD3 .4784E-07 .2169E-06 .2205 Masters (M) 155.21 147.88 1.050 M2 -.1136 .129 -.8782 M3 -.9865E-05 .2116E-04 -.4661 Doctoral (D) -122.65 270.65 -.4532 D2 .2422 .4377 D3 .18 77E-05 .2273E-0 3 .8255E-0 LD x UD .5256E-02 .4606E-02 1.094 .5532 LD X M .7186E-01 .5523E-01 1.301 LD X D -.7978E-01 .5703E-01 -1.399 UD x M .1322E-01 .1765E-01 .7491 UD x D .22 3SE-01 .2497E-01 .8963 M x D .5297E-01 .1655 .3201 * ** Significant at the ten percent level 5% 74 TABLE 6 (Continued) Variable Time Coefficient Estimate 32610.** Standard Error 3688. t-value 8.842** Central Michigan University Eastern Michigan University Ferris State College -15040. 190600 -.7895E-01 Lake Superior State College -140000. 101100 , -1.385 Michigan Techno­ logical University -59020. 47450 -1.244 -41580 72470, -.5737 128300.** 34810. 3.685** 445 70. .7932 Grand Valley State College Northern Michigan University Oakland University Saginaw Valley State College University of Michigan Wayne State University 35350. Western Michigan University 18520. 83700. -1.483 Constant -66630. 44930. -1.483 Significant at the ten percent level ie/ J-/0 75 TABLE 6 (Continued) Number of Observations: 192 Degrees of Freedom: 166 R2 .85239 R2 .830165 F-Tests for Groupings of Variables Linear Variables 2.758** Square Variables 1.192 Cubic Variables .8743 Interactions 3.252** Institutional Variables 2.588** ★ *■ Significant at the ten percent level 1 c/ X / O 76 TABLE 7 DATA CHARACTERISTICS— ENGINEERING Variable Lower Division(LD) Mean 1517 Standard Deviation 1667 LD 2 .5066E+0 7 .1347E+08 LD3 .2750E+11 .1317E+12 Upper Division (UD) 3948 3050 UD2 .2484E+08 .40 73E+08 UD3 .2128E+12 .5 734E+12 Masters (M) 628 586 M2 736500 .1469E+0 7 M3 .1233E+10 .4426E+10 Doctoral (D) 291 343 D2 201400 352400 D3 .1709E+09 .40 71E09 LD x UD .7922E+0 7 •1234E+08 LD M 788700 .1125E+07 LD x D 392200 623500 UD x M .3046E+0 7 .5199E+07 UD .1352E+07 .2 384E+0 7 320000 603600 M X D X X D $/LDSCH 100 MC AVC 20 X Max Min LDSCH 5,000 10,000 FIGURE 10 ENGINEERING LOWER DIVISION $/UDSCH 70 60 30 MC 20 AVC 10 X Min 20.000 10,000 0 -10 UDSCH X Max -20 FIGURE 11 ENGINEERING UPPER DIVISION $/MSCII 300 200 100 X Max X Min 0 1,000 2,000 -100 3,000 AVC MG FIGURE 12 ENGINEERING MASTERS MS CH $/D5CH /MG 1*00 300 200 AVC 100 Min X Max 600 800 1,000 -100 FIGURE 13 ENGINEERING DOCTORAL 1,200 1 ,1*00 * DSCH 81 variables; while only Lower Division Squared (-$.2328E-01) is significant and negative. Again it appears that the grouping of institutions is quite homogeneous for the entire state with the exception of the University of Michigan which is more expensive for the Engineering discipline as expected. Here, however, the first exception to the expectation of differing signs between comparable square and cubic terms is encountered at the Masters level although neither the square nor cubic term are significant at the 10% level. If both coefficients are actually negative, then there are increasing economies of scale. Examining Figures 10 through 13 shows again another combination of interactions and relationships. The Lower Division and Upper Division again show U-shaped curves but at the Masters level and Doctoral level the functions are nearly linear and the Masters level shows economies of scale while the Doctoral level illustrates diseconomies of scale at the margin within the relevant range. In this case all four student levels show portions of at least one of the cost curves falling below the horizontal axis in a variety of ways. For the first time it also appears that the square and cubic groupings of terms do not reach significance for the Engineering disciplines although both the individual t-tests at the Lower Division are significant. The other three groups of variables are significant for 82 Engineering. In fact these three groupings are significant for all six disciplines: Institutions. Linear, Interactions, and The Square and Cubic groups are significant for all disciplines with the exception of Engineering and the Social Sciences. Humanities The following variables are cost increasing and significant for the Humanities departments: Linear Lower Division ($24.65), Linear Masters ($138.60), Lower Division Cubed ($.6748E-08), Upper Division Cubed ($.8723E-07), Lower Division times Masters ($.2977E-02), Doctoral times Masters ($.2153), and Time ($52290). And the following variables cause costs to decrease and are significant: Lower Division Squared (-$.5100E-03), Upper Division Squared (-$.1790E-02), Upper Division times Masters (-$.6981E-02), Upper Division times Doctoral (-$.1481E-01) and all institutions with the exception of the University of Michigan and Western Michigan University. This gives reason to believe that the three institutions, the University of Michigan, Western Michigan University and Michigan State University are somewhat similar in their Humanities cost structure and that Michigan State is clearly among the more expensive in the state. TABLE 8 COST FUNCTION ESTIMATES- -HUMANITIES Variable Lower Division (LD) Coef f icient Estimate 24.65** Standard Error t-value 5.850 4.215** LD2 - .5100E-0 3* .2633E-03 -1.936* LD3 .6748E-08* .3018E-08 2.236* 11.55 1.203 .1035E-02 -1.730* .2435E-0 7 3.582** 59.01 2.349* Upper Division (UD) 13.89 - .1790E-02* UD2 •8723E-07** UD3 Masters (M) 138.59* M2 .7394E-02 .3233E-01 .2287 M3 -.7621E-05 .4769E-0 5 -1.598 106.33 -.3198 Doctoral (D) -34.00 D2 -.3366E-01 .1092 -.3083 D3 -.7974E-05 .2981E-04 -.2675 - .3080E-03 .1962E-0 3 -1.5 70 LD X UD LD X M LD X D -.1476E-02 .2152E-0 2 -.6859 UD X M - .6981E-02* .3354E-0 2 -2.081* UD X D -.1481E-01** .3687E-02 -4.016* .2573E-01 8.364* M X D .2977E-02** .2153** .1126E-02 Significant at the ten percent level 2. 644* 84 TABLE 8 (Continued) Variable Coefficient Estimate Standard Error Time 52290.** 2863. 18.26** Central Michigan University -154600.** 41600. -3.716** Eastern Michigan University -107000.* 43470. -2.462* Ferris State College -203200.** 48850. -4.160** Grand Valley State College -213800.** 49870. -4.288** Lake Superior State College -305500.** 107800. -2.833** Michigan Techno­ logical University -103800.* 60470. -1.717* Northern Michigan University -150900.** 44170. -3.417** Oakland University -133100.** 41700 -3 .192** Saginaw Valley State College -290500.** 79300. -3.663** University of Michigan -15210. 30590. -.4975 Wayne State University -57380* 31540. -1.819* Western Michigan University -24160. 36760. -.6572 Constant -95790.** 35460. -2.701** * ** Significant at the ten percent level 1% t-value 85 TABLE 8 (Continued) Number of Observations: 485 Degrees of Freedom: 454 R2 .87962 ■R2 .87166 F-Tests for Groupings of Variables Linear Variables 14.73** Square Variables 2.66** Cubic Variables 5.70** Interactions 3.67** Institutional Variables 12.31** * Significant at the ten percent level ** 1% 86 TABLE 9 DATA CHARACTERISTICS— HUMANITIES Variable Lower Division (LD) Mean Standard Deviation 12420 12860 LD2 .3193E+09 .6092E+09 LD3 .1141E+14 .2964E+14 7236 6484 Upper Division (UD) UD2 .9431E+08 .1483E+09 UD3 .1591E+13 .3475E+13 1052 1239 M2 .2638E+0 7 .5155E+0 7 M3 .8744E+10 •2294E+11 Masters (M) Doctoral (D) 388 693 D2 630200 .15 34E+0 7 D3 .1259E+10 .3612E+10 LD X UD .13608E+09 .2188E+09 LD X M .1822E+08 .3325E+08 LD X D .5858E+0 7 .132 7E+08 UD X M .1360E+03 .2293E+08 UD X D .4963E+07 .1160E+08 997400 .2394E+0 7 M X D $/ LDSCH MG AVC 20 00 X Max Min LDSCH 1 0 ,0 0 0 2 0 ,0 0 0 30,000 l |0 , 0 0 0 FIGURE 1 HUMANITIES LOWER DIVISION 50,000 60,000 $/UDSCH MG X Min AVC X Max 0 UDSCH 10,000 20,000 FIGURE 15 HUMANITIES UPPER DIVISION 30,000 $/MSCH 300 200 100 Min AVC^ X Max 1,000 2,000 5,000 3,000 \ \ ^MC FIGURE 16 HUMANITIES MASTERS MSGH $/DSCH 20 Min X Max 500 1,000 2,000 2,500 -20 AVG MO FIGURE 17 HUMANITIES DOCTORAL The five groups of variables all attain significance and the signs of the square and cubic terms differ for each student level with the exception of the Doctoral level; but just as in the case of the Masters level for Engineering, neither of the coefficients is significant where signs do not differ. The cost curves are U-shaped in the Humanities discipline for the Lower Division and the Upper Division although the Upper Division lies in the negative quadrant for most of the relevant range. The Masters level has a slightly inverted shape and the Doctoral level again shows an approximation of a linear relation with definite signs of economies of scale. Natural Sciences Table 10 presents the results for the Natural Science discipline where it is seen that the following variables are significant and increase total cost: Linear ($24.18), Linear Doctoral Lower Division ($383.40), Masters Squared ($.4344E-01), Doctoral Squared ($.1161), Upper Division times Masters ($ .2909E-01), Time ($37300), University of Michigan ($.1242E06), Wayne State University ($.8197E05), and Western Michigan University ($.1221E06). The variables which are significant and reduce costs are: Linear Masters (-$121.10), Upper Division Squared (-$.7121E-02), Masters Cubed (-$.3993E-05), Doctoral Cubed (-$.1168E-04), Lower Division times Masters 92 TABLE 10 COST FUNCTION ESTIMATES— NATURAL SCIENCES Variable Lower Division (LD) Coefficient Estimate 24.18** Standard Error 8.060 t-value 3.001** CM P P -.2847E-0 3 .4444E-0 3 -.6407 LD3 -.7138E-09 .6413E-08 -.1113 21.35 21.97 .9718 UD2 -.7121E-02* .3397E-0 2 -2.097* UD3 .1713E-06 .1625E-06 1.054 Upper Division (UD) Masters (M) -121.06* 51.07 -2.370* M2 .4344E-01* .2365E-01 1.837* M3 -•3993E-05* .1828E-05 2.184* Doctoral (D) 383.43** 67.27 5.700** D2 .1161* .4661E-01 2.490* D3 -.1168E-04** .4383E-05 -2.664** .1209E-02 .7478E-0 3 1.616 LD x M -.7926E-02** .2387E-02 -3.320** LD X D -.310 3E-02 .3271E-02 -.9486 UD X M .2909E-01** .8223E-02 3.538** UD X D .3429E-02 •1037E-01 .3304 .1846E-01 -5.818** LD M * ** UD X X D -.1074** Significant at the ten percent level 1% 93 TABLE 10 (Continued) Coeff i cient Estimate S.tandard Error t-value Time 37300.** 3033. 12.30** Central Michigan University- 53300. 48090. 1.108 Eastern Michigan University 58400. 44930. 1.300 Ferris State College - 110100 .* 46770. -2.354* Grand Valley State College -91570. 55840. -1.640 Lake Superior State College -147600.* 77440. -1.906* Michigan Techno­ logical University 21720. 41070. .5288 Northern Michigan University -25850. 45290. -.5708 Oakland University 125.7 47310. •2658E-02 Saginaw Valley State College -143600.* 84670. -1.696* University of Michigan 124200.** 28710. 4.328** Wayne State University 81970.* 34000. 2.411* 122100.** 43530. 2.805** -120800.** 36540. -3.306** Variable Western Michigan University Constant * ** Significant at the ten percent level 1% 94 TABLE 10 (Continued) Number of Observations: 399 Degrees of Freedom: 368 R2 .91287 R2 .90576 F-Tests for Groupings of Variables Linear Variables 13.26** Square Variables 2.106** Cubic Variables 4.109** Interactions 5.4.4** Institutional Variables 15.427** * Significant at the ten percent level ** 1% 95 TABLE 11 DATA CHARACTERISTICS— NATURAL SCIENCES Variable Lower Division (LD) Mean 10220 Standard. Deviation 10 77< LD2 •2202E+09 .4413E+09 LD3 •6 733E+13 .1865E+14 Upper Division (UD) 4476 3719 UD2 .3383E+08 .5433E+08 UD3 .3409E+12 .8102E+12 Masters (M) 1052 1442 M2 .3182E+0 7 .8888E+07 M3 .1508E+11 .6195E+11 Doctoral (D) 601 1030 D2 .1418E+0 7 .4122E+07 D3 .4687E+08 .2042E11 LD x UD .7087E+08 .1212E+09 LD x M .1808E+08 •4042E+08 LD x D .1163E+08 .2925E+08 UD x M .9219E+0 7 .20 20E+08 UD x D .5643E+0 7 .136 2E+08 M .1331E+07 .5545E+0 7 X D $/LDSCH 20 10 1,000 2,000 AVC 5,000 3,000 \ LDSGH Min X Max MC FIGURE 18 NATURAL SCIENCES LOWER DIVISION $/UDSCH MC vO ~v3 X Max Min 10,000 15,oop 20,000 / 25,000 UDSCH AVC FIGURE 19 NATURAL SCIENCES UPPER DIVISION $/MSCH 100 .VC X Max J MSCH (5,000 2,000 -20 MC -60 -80 FIGURE 20 NATURAL SCIENCES MASTERS $/DSCH 800 700 500 I]0 0 AVC MC 300 200 X Max 1,000 2,000 3,000 it,0 0 0 FIGURE 21 NATURAL SCIENCES DOCTORAL 5 ,0 0 0 6,000 100 (-$.7926E-02), Masters times Doctoral (-$.1074), Ferris State College (-$.1101E06), College (-.1436E06). and Saginaw Valley State Again each of the five groups of variables is significant. At each level except the Lower Division the sign of the Cubic and Square terms differs and at the Lower Division, neither the Square nor the Cubic term is significant at the 10 percent level. For the Natural Sciences, the Lower Division cost curves approximate linear functions, the Upper Division takes the U-shape and the graduate levels are both inverted U-shaped. The Upper Division curves have negative values for a surprising portion of their range, the Masters level curves begin in negative territory but rapidly become positive and the Doctoral curves and Lower Division curves are universally in the positive quadrant. These can be seen in Figures 18 through 21. Social Sciences The results for the Social Science discipline are found in Table 12. increase costs are: Variables which are significant and Upper Division ($20.73), Doctoral Linear ($121), Upper Division Cubed ($.2558E-07), Upper Division times Doctoral ($.6684E-02), Time ($27020), and University of Michigan ($65450). Those variables which are significant and reduce costs are: Upper Division 101 TABLE 12 COST FUNCTION ESTIMATES— SOCIAL SCIENCES Variable Lower Division (LD) Coefficient Estimates 1.072 Standard Error 6.176 t-value .1736 LD 2 .2091E-03 .4289E-0 3 .4875 LD3 - .1501E-08 .7536E-08 -.1992 Upper Division (UD) 20.73* 8.095 2.560* UD2 -.1154E-02* .6320E-03 -1.826* UD3 .2558E-07* .1539E-0 7 1.663* Masters (M) 27.43 25.07 1.094 M2 .3098E-02 .5147E-02 .6020 M3 - .1516E-06 .2508E-06 -.6046 Doctoral (D) 121.04* 55.02 2 .2 0 0 * D2 - .1500E-01 .1840E-01 -.8153 D3 .1593E-05 .2064E-05 .7716 LD x UD .2298E-03 .3932E-0 3 .5844 LD x M .1513E-03 .2077E-02 .7283E-01 LD x D .1188E-02 .2187E-02 .5434 UD x M .6014E-03 .2077E-02 .2895 UD x D .6684E-02** .2026E-02 3.300** 3874E-01** .1147E-01 -3.377** M * ** X D Significant at the ten percent level 1% 102 TABLE 12 (Continued) Coefficient Estimate Standard Error t-value Time 27020.** 2160. 12.51** Central Michigan University -51790.* 29420. -1.760* Eastern Michigan University -35430. 32060. -1.105 Ferris State College -95327.** 31390. -3.036** Grand Valley State College -36660. 39860. -.9197 Lake Superior State College -167800.** 62530. -2.684** Michigan Techno­ logical University -157700.* 69340. -2.274* Northern Michigan University -71070.* 28620. -2.483* Oakland University -65830.* 32520. -2.024* Saginaw Valley State College -162000.** 57910. -2.797** University of Michigan 65450.** 22070. 2.966** Wayne State University 26160. 20970. 1.248 Western Michigan University -33134.8 25720. -1.288 Constant -25440. 27110. -.9385 Variable Significant at the ten percent level 103 TABLE 12 (Continued) Number ofObservations: 344 Degrees 313 of Freedom: R2 .92769 R2 .92076 F-Tests for Groupings of Variables Linear Variables 6.190** Square Variables .9957 Cubic Variables .7792 Interactions 3.434** Institutional Variables 7.944** * Significant at the ten percent level ** 1 % 104 TABLE 13 DATA CHARACTERISTICS— SOCIAL SCIENCES Variable Lower Division (LD) Mean 8512 Standard Deviation 7550 LD2 .1293E+09 .2526E+09 LD3 .2874E+13 .8928E+13 Upper Division (UD) 7255 6082 UD2 .8951E+08 •1476E+09 UD3 .1487E+13 .3 711E+13 Masters (M) 1413 1717 M2 •4937E+07 .186 3E+08 M3 .3404E+11 .2586E+12 Doctoral (D) 606 1136 D2 .1654E+0 7 .5659E+0 7 D3 •6754E+10 .3472E+11 .8698E+08 .1321E+09 LD x M .1 349E+08 .2093E+08 LD x D .8820E+0 7 .2265E+08 UD x M .1445E+08 .2 285E+08 UD x D .9014E+0 7 .2282E+08 M •1575E+0 7 .3765E+07 LD UD X X D $/LDSCH AVC 10 Min X Max 20,000 60,000 FIGURE 22 SOCIAL SCIENCES LOWER DIVISION 105 MC $/UDSCI] MC AVC 20 106 10 X Max Min -UDSCH 10,000 20,000 FIGURE 23 SOCIAL SCIENCES UPPER DIVISION 30,000 $/MSCH 35 30 AVC 20 MG 15 10 5 X X Max MSCH 0 5 ,ooo FIGURE 2i| SOCIAL SCIENCES MASTERS 10,000 $/DSCH 125 MG AVC 100 108 X Max Min J 1,000 2,000 3,000 1 |,0 0 0 FIGURE 25 SOCIAL SCIENCES DOCTORAL 5,000 6,000 DSCH 7,000 109 Squared (-$.1554E-02), Masters times Doctoral (-$.3874E-01), Central Michigan University (-$.5180E05), Lake Superior State College (-$.1678E06), Michigan Technological University (-$.1577E06), Northern Michigan University (-$.7107E05), Oakland University (-$.6583E05), and Saginaw Valley State College (-$.1620E06). Social Sciences and Engineering are the only two disciplines for which all five groups of variables are not significant as noted earlier and in both cases the Square and Cubic groups fail to gain significance at the ten percent level. In both cases the Square group and the Cubic group have one significant variable each, using individual t-tests. For the Social Sciences both were at the Upper Division and for Engineering both were at the Lower Division. Again the signs of the square and cubic terms differ at all levels. As can be seen in Figures 22 through 25, the Upper Division and Doctoral levels exhibit U-shaped cost curves while the Lower Division and Masters levels show inverted U-shapes. The Social Sciences cost curves are universally in the positive quadrant for the only time. Summary and Comparisons Table 14 summarizes the sign and significance results for the six disciplines estimated. 110 TABLE 14 SIGNS AND SIGNIFICANCE OF THE COEFFICIENTS Variable Busi­ ness LD Education Engineering +* +* UD +* +* +* M _★ +* + D +* + _* LD2 UD2 _* M2 D2 — LD3 * Nat, Sci, +* +* + + +* _★ + +* +* +* _* + + _* _ * +* +* + + +* +* +* UD3 + _* + +* M3 + D3 +* ★ + _ LD X M +* +* LD X D _* _ 'k UD X M +* UD x D M D Time Constant _* +* + _* LD x UD X Soc. Sci. Humanities + _* + + + + + +* _* + +* + _* +* + + _* + _* + +* _* + + +* _* _* +* +* +* +* +* +* _* _* Significant at the ten percent level Ill TABLE 14 (Continued) Variable Business Education Engineering Humanities Nat. Sci. Central Michigan Soc. Sci . _ * Eastern Michigan Ferris State -* + - -* -* _* _* j. _ _* _ * _-k Grand Valley Lake Superior Michigan Tech Northern Michigan -* -* Oakland -* + - + + - -* _* -* _* -* _* Saginaw Valley _* Univ. of Michigan +* Wayne State +* +* - +* +* + + + -* +* + Western Michigan -* + + - +* Linear Terms * * * * * Square Terms * * * * Cubic Terms * * * * * * Inter­ actions Institu­ tions * ★ * * * * * 112 The linear terms at the undergraduate level have universally positive coefficients and eight of the twelve coefficients are significant; and at the graduate level, four estimates are negative with two attaining significance and eight are positive of which five are significant. Fifteen of the twenty-four total linear variables are significant. The fact that two negative estimates reach significance at the graduate level was not expected but can probably be explained by the facts that 1 ) under­ graduate programs are ordinarily large before graduate programs are introduced and there were no purely graduate programs in the sample used and 2 ) graduate students are clearly used as inputs into the undergraduate educational process. The savings incurred by introducing a graduate program may thus show up either in the interaction term or in any of the other terms since there are few low undergraduate output data points with graduate programs present. Here a portion of the savings realized has shown up in the linear term at the Masters level for the Business Administration and Natural Sciences disciplines. It was expected, ci priori that the Natural Science discipline would show one of the strongest levels of savings with the introduction of a graduate program because of the use of graduate assistants in the laboratory for instructional purposes as well as research. The square terms are negative in sixteen cases and positive in eight with nine significant and negative and 113 three significant and positive. The total of twelve of twenty-four estimates reaching significance again indicates that the square term is important and that there clearly are non-linear relationships extant. At the undergraduate level only two of the eight estimates are positive, neither of which is significant, and seven of the ten negative estimates are significant. At the graduate level on the other hand half of the estimates are positive with three significant and positive and two significant and negative. The cubic terms have fourteen positive estimates with seven significant and ten negative of which three are significant. At the undergraduate level, three of the twelve estimates are negative while one of the six significant estimates is negative. The graduate level estimates are again five positive and seven negative with two positive and significant and two negative and significant. As noted earlier the relationship between the square and cubic terms has turned out as expected with the cubic coefficient estimate significantly smaller than the square coefficient and with signs differing for the two terns for a given output level within a discipline in all but four instances of twenty-four. In each instance neither of the terms was significant if signs did not differ. It should be noted that of the five groupings of variables, only the square and cubic groupings fail to 114 show significance; instances: and that these each fail in two Engineering and the Social Sciences. The interaction terms show of which eight are significant, which ten are significant. 21 positive estimates, and fifteen negative, of The fact that half of the terms in this category are significant again indicates its importance. For each of the interaction terms except Lower Division times Doctoral, there is a mixture of signs and where more than one discipline shows significance for the variable there is at least one positive and one negative and significant term. At the Lower Division times Doctoral all estimates are negative and two are significant. The only level where there is only one discipline showing significance is Lower Division times Upper Division where Education shows a negative significant relation but four of the six estimates are positive though not significant. There are no very clear consistencies among the six disciplines tested with respect to the interactions found but none were expected. The Time Trend variable on the other hand shows complete consistency with all estimates positive and significant as was expected. The smallest t-value encountered for this variable was 6.734 for Education and the largest was 13.26 for the Humanities. This variable was very clearly of significance. The constant term, included for statistical reasons as it was not expected to be significant as a representative 115 of fixed instructional costs, was negative in all disciplines and significant for two. This term will represent the mean effect of left-out variables in addition to its other roles, and the fact that it is universally negative and significant for two of the six disciplines makes interpretation difficult unless the left-out variables are cost saving which may be the case but seems unlikely. The institutional dummy variables were positive in twenty-six instances, negative in forty instances, and significant in a total of thirty-six cases with twenty-nine significant and negative and seven significant and positive. Of the seven which are significant and positive, the University of Michigan accounts for five; only in one instance did the University of Michigan not have significantly higher costs than Michigan State University and that was for the Humanities where all institutional dummy variables had negative coefficients and ten of the twelve were significant and negative. The only two significant positive institutional dummy variables which were not from the University of Michigan were in the Natural Sciences for Wayne State University and Western Michigan University. The entire picture shown in this research by the institutional dummy variables fits well with expectations regarding the distribution of signs among the disciplines and institutions for this variable. 116 This section has presented the results of the estimation of the cost function for instruction in higher education for the six disciplines chosen for this study. The model presented in Chapter II has fit well with expectations, has shown each of its terms to be significant for at least one discipline, and has been shown to be highly predictive. The only apparent anomaly is found in the constant term where it was expected that there would be no fixed costs included and where it was expected that the mean effect of any left-out variables would be found; and yet the constant has been found to be universally negative and twice significant and negative. Tests of Hypotheses The five hypotheses to be tested in this section fall into two groups: 1 ) those related to the assumptions necessary in order to use present cost accounting techniques to discover either marginal cost or average variable cost associated with one of the outputs of higher education and 2 ) those implied by the assumptions of the model used herein and the results of present costing techniques. separately. These two groups will be dealt with 117 Present Costing Assumptions As noted earlier, the presently used cost accounting procedures identify the prices of those resources directly used in the production of a given output, add these to find the total cost of an output, and then divide by total output in order to find the average or unit cost of a given output. If the prices of inputs to a given output are a function of the quantity produced of another output, or if the production technique utilized is 4 function of the presence of a resource implied by the presence of another output, then this procedure does not show the true relation, more correctly it may not show the true relation, among costs. If there are economies or diseconomies of scale in production then the cost figures produced by such a procedure will not also represent marginal costs except in very unusual circum­ stances though they may be successful in estimating average variable costs. If there is a possibility that within some range of outputs negative marginal costs may be encountered due to the presence of an interaction among the levels of output and therefore that there is a possibility that average variable costs are in reality negative within some ranges which may be encountered, present cost accounting techniques will not discover these negative cost ranges since the technique itself implies positive estimates of unit costs because AVC is 118 by definition here the ratio of two positive quantities. The above implies that in order to use current cost accounting procedures the following assumptions must be made: 1) there are no interactions among levels of instructional output with respect to costs, 2) there are no economies or diseconomies of scale encountered in the production of instruction, and 3) there are no negative average variable costs encountered within the range of outputs which actually occur. From these assumptions, the following hypotheses in the negative form of the above are to be now tested: 1) the inter­ action terms of the estimated cost function for instruction in higher education are significant, 2 ) the nonlinear terms of the estimated cost function are significant, and 3) the average variable costs when calculated using the estimated cost function are not significantly greater than zero in all instances and are significantly less than zero in some instances. None of these results is implied by the assumptions of the model but it is clear that interactions are possible, a non­ linear relationship is possible, and average variable costs may in fact be negative within certain ranges. The results necessary to test hypotheses one and two have already been presented above. The results necessary to test hypothesis three will be developed and presented below. In each case, implied by the model, since no sign or magnitude is two tailed tests will be used at 119 the ten percent level of significance. Hypothesis One There were six interaction terms included in the estimate of each of the six disciplinary instructional cost functions. Of the total of thirty-six estimates, eighteen were significant, twenty were positive and sixteen were negative with ten negative and significant and eight positive and significant. The interaction between Lower Division and Upper Division had only one discipline where it was significant, Education; Lower Division times Doctoral was significant for two disciplines, Business and Education; Upper Division times Doctoral was significant for Education, Humanities, and Social Sciences; and the remaining three interaction terms were significant for four disciplines with all reaching significance for Business, Humanities, and Natural Science; Lower Division times Masters and Upper Division times Masters attaining significance for Education; and Masters times Doctoral was significant for Social Science. From a disciplinary standpoint, individual tests on the variables showed that none were significant for Engineering, two for the Social Sciences, three for the Natural Sciences, four for Business Administration and the Humanities, and five for Education. In no case did 120 a pattern of significant signs emerge for any discipline. Only in the case of Lower Division times Doctoral, where all estimates were negative and two of these were significant, did a variable show any consistency in signs. The empirical results clearly indicate however that there are statistically significant interactions among levels in the instructional cost function though these do not show a consistent pattern. The inconsistent pattern is not contrary to expectations, however, since the interaction was expected to depend upon instructional technology and acceptable pedagogy which clearly vary among disciplines. The conclusion reached therefore is that there are in fact interactions among levels which do affect costs. Hypothesis Two As analyzed above, the non-linear terms also clearly appear to be significant as twelve of the square terms attain significance and ten of the cubic terms are significant. Unlike the interaction terms, the non-linear terms appear to show a clear pattern among the signs of the estimates although the pattern is not perfect. The Pattern: Of the twenty-four combinations of instructional level, discipline, and square-cubic terms; in only four cases do the signs of the cubic and square 121 term for the estimates agree: Masters and Doctoral for Engineering, Doctoral for the Humanities, and Lower Division for Natural Sciences. In the other twenty cases, the sign of the square term differs from the sign of the cubic term. None of the eight estimates involved in same signs for both non-linear terms was significant. This differing sign result was expected since economies of scale were expected to disappear as output approached the point where maximum class sizes would be reached for all sections. And diseconomies of scale were expected to disappear as programs became large enough so that faculty suited to higher levels of instruction would be able to specialize at the maximum level of their qualification. These same factors were expected to produce economies of scale, if there were any nonlinearities, at the undergraduate level and indeterminate effects at the graduate level; and this, in fact, is what the analysis has shown. At the undergraduate level, the sign of the square term, the term which determines initial economies or diseconomies of scale, is negative in ten of twelve instances, significant and negative in seven instances, and significant and positive in no cases. At the graduate level on the other hand, the sign of the square term varies, being positive in six cases and negative in six cases with three cases of positive, significant results and two cases of negative, significant results. 122 These results confirm the existence of significant initial economies and diseconomies of scale in instruction. The cubic terms show the same regularity of sign at the undergraduate levels with nine of twelve estimates positive and three negative, five estimates positive and significant, and only one negative and significant. The graduate level again shows variability. It also appears strongly supported that there is a tendency for initial (dis)economies to disappear as programs become larger. There are nine instances in which both the square and cubic term are significant for a level-discipline, and in each case the signs differ between the two terms. Seven of these nine cases show initial economies of scale with five of these seven at the undergraduate level: Lower Division for Education, Engineering and the Humanities, and Upper Division for the Humanities and Social Sciences. There are no cases in which there appear to be initial diseconomies of scale at the undergraduate level where both the square and cubic terms are significant. At the graduate level there are four cases where both terms are significant: Masters in Education and Doctoral in Business Administration show initial economies while Masters and Doctoral in Natural Sciences both show initial diseconomies. The conclusion reached is that there are significant 123 non-linearities in the cost function, that these tend to be economies of scale at the undergraduate level and may be either economies of scale or diseconomies at the graduate level, and that these non-linearities are extant initially but have a tendency to disappear at higher levels of production. Hypothesis Three In order to test the hypothesis that there are negative average variable costs of instruction it will be necessary to provide further information: the average variable cost of the outputs, and the standard error and t-statistic. The Average Variable cost of the four outputs will be defined as follows: AVC LD = B lX l+ B 5X l+B9X l+B25X'lX 2+B25X lX 3+B27X lX 4 ~~\r 1 AVC... UD = B 2X 2+B6X 2+B10X 2+ _ B 25X __ 1X 2+ B 28X 2X 3+B29X 2X 4 2 AVC.. M = B 3X 3+ B 7X 3+B11X 3+B2SX 1X 3+B28X 2X 3+ B 30X 3X 4 -------------------- x ----------------------3 AVC = B4X 4+B8X 4+B12X 4+B2 7X 1X 4+B29X 2X 4+ B 30X 3X4 X4 These are simply those terms in which the output appears, those which can be affected by changes in that output or those which are variable, divided by output at the level 124 in question. These expressions can be simplified by dividing through by the denominator in each case since the denominator appears in each term of the numerator but they are shown in this form in order to clarify the definition. The simplified form would of course not be defined at zero for the output in question. The average variable costs by level and by discipline computed at the mean output for each student level are found in Table 15 with the standard error and t-value. The point represented by the mean output for each of the four levels is chosen to evaluate the above formulae because these are points that are in the center of the data for each discipline, where confidence should be the greatest regarding the parameter estimates and because a point must be chosen. The standard errors are computed using the standard formula for computing this statistic for a linear combination of random variables and values obtained from the variance covariance matrix of the coefficients. Essentially the output values are assumed fixed at the mean output levels for each of the four outputs and the parameter estimates are treated as random variables with estimated variance and covariance values. In Table 15, it can be seen that seventeen of the values are significant of the twenty-four and that two are negative and significant and fifteen are positive and significant. The two negative significant values 125 TABLE 15 AVERAGE VARIABLE COSTS Discipline Level Average Cost S-tandard Error t-value Business Administration LD 8.61 2.353* 45.50* 17.32 2.627* M -7.90 7.81 D 491.74* 203.02 2.422* .2896 UD 20.26* -1.011 Education LD 8.13 28.07 UD 20.31* 4.68 4.336* M 62.45* 9.79 6.382* D -9.95 48.96 LD 67.96* 37.67 1.804* UD 51.27* 13.48 3.803* M 256.65* 95.40 2.690* D -51.42 154.09 -.2032 Engineering -.3338 Humanities LD 19.SS* 1.89 10.413* UD -11.39* 6.79 -1.678* M D 207.94* 52.70 34.98 5.945* 76.81 .6861 TABLE 15 (Continued) Discipline Level Average Cost Standard Error t-value Natural Sciences LD 16.42* 4.37 3.757* UD 37.83* 15.47 2.445* M -28.51* 15.87 -1.796* D 319.64* 54.11 5.907 Social Sciences * LD 5.30 3.79 1.397 UD 20.55* 5 .49 3.742* M 13.67 15.91 .7552 D 116.4* 36.67 3.174* Significant at the ten percent level 127 occur in the Humanities at the Upper Division and in the Natural Sciences at the Masters level. Of the seven values that fail to attain significance, three are negative and four are positive. Two of these negative non­ significant values occur at the Doctoral level, Education and Engineering, and one is at the Masters level, Business Administration. The only surprise here is the negative and significant average variable cost at the Upper Division in the Humanities. This must be taken to indicate that the Humanities departments use a large number of upper division students to aid in the instructional process. It is also possible that teaching upper division students is sufficiently more desirable that lower faculty salaries per student can be paid by departments with upper division programs. The first of these possibilities is made more likely by the relatively high average variable cost encountered for the Humanities at the Masters level. The high Masters level figure, relative to other disciplines, could be explained by the fact that much of the possible student assistance that can be accomplished by non-doctoral level students is already done by upper division students before masters programs are introduced. Even if after masters programs are introduced and these student assistance duties are taken over by the graduate students from the upper division students, much less would be saved in this way than would be saved if the graduate assistant were 128 replacing a faculty member; and this option may actually be more expensive but necessary in order to recruit graduate students. 20 This would also be consistent with the relatively low estimate of average cost at the Doctoral level. The second of the above possible explanations seems relatively unlikely but possible. There is of course the third possibility that this is not really a parameter that is less than zero. Nine of the average variable cost estimates are not significantly greater than zero and two of these are both negative and significant. This indicates that it is certainly possible that some negative average variable costs may be encountered at the mean output levels, but it certainly cannot be considered to be conclusive evidence. The evidence is not much stronger if the graduate level values alone are examined. The graduate level is the level at which it would be expected that the effects which may lead to negative average variable costs would be most likely to occur. Although four of the twelve estimates are negative and one of these is significant, the strongest evidence for the possibility of negative average variable costs occurs in the Natural Sciences. Here it would be expected a^ priori that the cost saving effects of a graduate program would be the greatest due to the extensive faculty time required in laboratories which can be replaced by high quality graduate students at much lower cost. And the masters 129 level coefficient is significant and negative. With respect to this hypothesis, the conclusions reached must be more cautious than those of the first two hypotheses, however. It does appear that negative average variable cost estimates are clearly not out of the question. Therefore, some consideration should be given to a costing technique which aims to estimate average costs but does not eliminate the possibility of negative average cost results by definition. Model Generated Hypotheses Two hypotheses were generated from the model and presently accepted cost data. These are: 1) Marginal costs are universally greater than zero and costs increase with student level. 2) marginal The tests of both of these hypotheses require additional data. Positive Marginal Costs Table 16 presents the marginal cost values as evaluated again at the mean output value for each output. The marginal cost at each level is computed by taking the first partial derivative with respect to each level of output respectively which results in the following: 130 TABLE 16 M/RGINAL COSTS Discipline Level Marginal Cost Standard Error t-value Business Administration 5.01 3.253* 24.75 34.96 .70 79 M -15.03 9.47 -1.587 D 188.06 124.44 1.471 -12.45 23.63 .5269 LD 16.31* UD Education LD UD 21.53* 3.77 5.713* M 34.64* 5.90 5.874* D 46.37 43.88 1.057 LD 39.46* 15.78 2.520* UD 40.70* 6.99 5.818* M 177.52* 57.86 3.068* D 19.29 94.81 .2034 Engineering Humanities LD 15.43* 1.68 9.205* UD -15.21* 3.77 -4.031* M 198.86* 22.70 8.760* D 37.24 66.16 .5629 TABLE 16 (Continued) Marginal Cost Standard Error LD 13.36* 2.070 6.453* UD 12.82 8.68 1.478 8.35 23.66 .3529 36.54 10.427* Discipline Level t-value Natural Sciences M D 380.96* Social Sciences * LD 6 .86 * 2.10 3.260* UD 14.87* 3.42 4.342 M 17.44 10.99 1.586 D 108.49* 28.16 3.853* Significant at the ten percent level 132 MCLD = B1+2B5X 1+3B9X ?+B25X 2+ B 26X 3+B27X 4 MC MC M B 3+2B7X 3+3B11X 3+B26X 1+ B 28X 2+B30X 4 D The table presents the marginal cost figures in the first column, the standard error in the second column and the t-value in the third column. Again the standard error is computed using the standard formula for linear combinations of random variables using the variance covariance matrix of the coefficients. Comparing the results found in Table 16 with those in Table 15, it should be noted that the results are the same with respect to significance and sign with the following exceptions: Upper Division and Masters level average costs are both significant for Business Adminis­ tration but the marginal cost estimates, while both positive again, do not attain significance at the ten percent level; there is a sign shift in Education at the Lower Division and Doctoral levels although none of these estimates is significant; a sign shift also takes place at the Doctoral level in Engineering; in the Natural Sciences the Upper Division significance is lost in the marginal cost table and a negative significant estimate of marginal cost at the Masters level is positive and non­ 133 significant for Marginal cost; for the Social Sciences the only difference lies in the gaining of significance at the lower division. Two AVC estimates were negative and significant but only one of these, Humanities at the Upper Division remains negative and significant as the focus changes from AVC to MC. And five negative estimates of AVC can be compared to only three for MC. Once again there is evidence of the possibility of negative marginal costs but the evidence is certainly not conclusive. Marginal Costs Increase with Student Level The data necessary to test this hypothesis are found in Table 17. These are presented such that the relevant differences between marginal costs for a discipline as student level increases are shown in column one, the relevant standard error is shown in column two, and the t-value is shown in column three. The movements in marginal costs are measured in the three steps: 2) 1) Lower Division to Upper Division, Upper Division to Masters level, and level to Doctoral level. 3) Masters These were chosen because there has been clear evidence that costs increase from one level to another for averaged departments and for individual departments with very few exceptions. The results show that of the eighteen comparisons, 134 TABLE 17 DIFFERENCES BETWEEN MARGINAL COSTS Discipline Level Difference Difference Standard Error t-value Business Administration UD - LD 8.44 36.14 .2335 M - UD -39.78 34.97 -1.138 D - M 251.93 121.44 1.631 UD - LD 3 3.98 24.62 1.380 M - UD 13.11 8.43 1.554 9.73 44.35 .2194 1.23 19.33 .0636 136.81* 61.44 2.227* -158.23* 60.57 -2.612* UD - LD -30.64* 5.51 -5.561 M - UD 214.15* 20.91 10.24* -166.58* 77.74 2.078* Education D - M Engineering UD - LD M - UD D - M Humanities D - M Natural Sciences UD - LD -.54 9.20 -.0587 M - UD -4.57 53.49 -.0854 D - M 372.61* 40.43 9.216* 135 TABLE 17 (Continued) Discipline Level Difference Difference Standard Error t-value Social Sciences UD - LD 8.01 M - UD 2.57 D - M * 91.05* 6.41 1.250 9.75 .2636 30.03 3.032* Significant at the ten percent level 136 seven are significant and, of these, four are positive and three are negative. Of the eleven estimates which do not reach significance, eight are positive and three are negative. Only two of the disciplines show consistent rises in marginal costs as student level rises: Education and Social Science. The individual increases for these two disciplines are not significant except for the Masters to Doctoral increase for the Social Sciences if two-tailed tests are used. The use of one-tailed tests which should have been implied a priori by former evidence would result in two of the Education differences and one Business Administration difference attaining significance. The fact that so many of the estimates are significant and negative if two-tailed tests are used seems to make the use of a one-tailed test out of the question. One-tailed tests would affect only three of the estimates: two in Education and one in Business Admini stration. In the case of the Natural Sciences, only the positive shift in costs from the Masters to Doctoral level is significant. For Engineering, two of the differences are significant with one negative and one positive in each case. The Humanities show three significant differences with two negative and one positive. There is only one discipline which shows a significant difference at the Lower Division to Upper Division level, Humanities, where this difference is negative. The 137 Upper Division to Masters level is significant for two disciplines: Humanities. positive for both Engineering and the The Masters to Doctoral difference is significant in four of the six disciplines but the sign is positive in two cases and negative in two. The only disciplines where the Masters to Doctoral difference is not significant are Business Administration and Education where the estimates are both positive. The conclusion reached is that there is clear evidence that precludes the acceptance of the hypothesis that all marginal costs increase with increases in student level at least at mean output levels. A one-tailed test would show positive and significant results for eight of the eighteen possible cases, but the large number of negative estimates and three cases of two-tailed significant negative results makes acceptance of the hypothesis impossible. Summary It is clear from the results of the estimation procedure that the cost function estimated yields significant predictive power; however, the tests of the two hypotheses implied by the model and expectations of decision makers with reference to relative costs based on previously available data do not yield supportive results. This is consistent with the results of the checks 138 on the assumptions under which the previously available data were produced. In two cases, the assumptions were rejected and in the third, there was significant evidence to reject. The apparent non-validity of the assumptions under which previous data were produced apparently led to inaccurate cost estimates which in turn precluded decision makers from maximizing welfare as indicated by the results of the tests of the final two hypotheses. Unrealistic assumptions do not necessarily imply results which do not correspond to reality; but in the case of the costing procedure used to estimate costs of instruction in the State of Michigan at least, these unrealistic assumptions seem to have led to conclusions which may have led to non-optimal allocation of resources in at least the two disciplines where marginal costs have significant declines as student level increases: Engineering and the Humanities. The assumptions that outputs at the various student levels are independent, that there are no (dis)economies of scale and that negative costs are not possible have been un­ realistic enough to result in possible nonattainment of the welfare maximum. Thus there have been arguments presented which should demonstrate the inadequacy of present procedures and the inaccuracy of present conceptions of the cost relationship to those who judge a model on the basis of the realism of the assumptions and to those who judge a model on the basis of the validity of the results. 139 Footnotes Chapter III 18. Unit Cost Study. 19. The t-values are high enough that such questions do not arise. If this were not the case, one- tail tests would clearly be appropriate. 20. It is assumed here that graduate students would be deemed to provide superior service though undergraduates would suffice. CHAPTER IV Summary and Conclusions A model of the productive process in higher education was presented in Chapter II which generated the nature of the total cost function for a department. The hypothesized form of the total cost function was then estimated and tested in Chapter III. The results presented in Chapter III clearly show that this model has significant predictive power. All of the variables are significant in at least one of the six disciplines examined as can be seen in Table 3.8. In general there is not very much consistency among the disciplines tested except in the following ways: 1) in each case the time trend is positive and significant, 2) the estimates of the linear terms are all positive and many are significant at the undergraduate level, 3) some non-linearities are present for each discipline with a strong tendency toward economies of scale at the under­ graduate level and a mixture of economies and diseconomies at the graduate level, 4) all non-linearities have a tendency to disappear at higher levels of output where individual coefficients are significant and in all but a few cases where the variables do not attain significance, 5) each of the disciplines except Engineering shows at least two significant interactions among the student levels 140 141 of instructional output but here there is no clear pattern of regularity among the disciplines with reference to signs, 6) each of the institutional dummy variables is significant in at least two of the disciplines although in the cases of Education and Engineering, only the University of Michigan shows as being significant; the signs of the significant variables correcting for institutional differences are as expected, positive for the University of Michigan and negative for other institutions with the exception of two cases, Wayne State University and Western Michigan University in the case of the Natural Sciences, and 7) in each case in which the groups of variables were tested they were found significant with the exception of the square and cubic terms in the cases of Engineering and Social Sciences where in each case there was evidence of non-linearity at one level in both the square and cubic terms. The tests of the hypotheses generated by the assumptions necessary for the use of present costing procedures clearly demonstrate that these assumptions are not met and the hypotheses based upon the results of the present costing procedures and the model of Chapter II clearly show that the results of the procedures have been erroneous and have led to failure to maximize welfare unless the assumptions under which the model of Chapter II is built are not met. The policy implications of these results are 142 potentially significant. These include the desirability of maintaining a data system which not only records the instructional output of the system of higher education by department but also records all of the output in a systematic and consistent manner. There are clearly difficulties involved in maintaining such information; however, if decisions are to be made which will maximize the benefit of the activities of the system of higher education provided by the various governmental agencies, such data are mandatory. One of the greatest drawbacks of the present study was the inability to account in a quantitative manner for research and public service produced as a result of the expenditures in the instructional budget. It is felt that had accurate data been available on these outputs the results involving negative marginal and average variable costs would have been clearer since it is expected that both research and public service activity would be greatest at graduate level institutions where it would have a tendency to lower the costs of graduate level output. The fact that the constant term was found to be negative for all disciplines and significant for two disciplines, may indicate that these left-out variables— they were included as qualitative variables— may have had the opposite effect of reducing costs and increasing the coefficients of the instructional outputs and particularly at the graduate level. 143 There may well be arguments now for increasing the size of the average department in all disciplines since wherever economies of scale were present, almost universally at the undergraduate level, these were not yet used up at the average departmental size. This implication is not necessary, however, because there are other considerations affecting community welfare with respect to location of educational plant and costs in terms of foregone income and travel time, etc. It seems clear that upper division programs in the Humanities should be encouraged wherever there is a demand for them since upper division programs seem to lower total costs. The same can be said for masters programs in the Natural Sciences. If in fact instructional costs can be saved at the same time that additional opportunities can be offered to a local community, it would seem to be non-rational if such services were not provided if there were a demand. The argument in the case of the Natural Sciences may not be as strong as in the case of the Humanities since there may well be expenses incurred in the production of specialized laboratory facilities for masters level programs in certain departments which would more than outweigh the savings realized in instruction. Unfortunately the cost of such specialized instructional facilities is not included in the instructional budget. Since some areas 144 in the Physical Sciences are experiencing a slackening in demand for graduate level programs, the argument for eliminating these was stronger than it is now that the possibility of net savings as a result of masters level programs has been shown. Each department is unique and must be viewed in that way: 1) each department will have a different mix of faculty by rank, 2) each department will have a different mix of tenured and non­ tenured faculty, 3) each department has a different set of specialized physical facilities available to it, etc. The analysis of both marginal and average costs and the differentials in marginal costs may have other implications with regard to the mix of programs by level. It is clear in the case of negative costs that programs should be encouraged, and it is equally clear that if decision makers felt the mix of outputs to be optimal under the assumption that costs increase with student level these same decision makers should now feel that the mix of outputs is non-optimal when it is seen that the relative costs may well not be as they were previously perceived. In the case of Business Administration disciplines, it would seem more likely that masters level programs would be encouraged when it is noted that masters level programs are potentially less expensive than upper division programs in the same discipline. The same should be true of doctoral level programs in Engineering and the Humanities. If at present output the marginal student 145 in these disciplines at the doctoral level was thought to have a value in excess of the marginal masters student and it is now discovered that the doctoral student is much less expensive than the masters student, only two responses seem appropriate: from present levels, or 1) increase doctoral output 2) decrease masters level output. So changes in departmental size, departmental location, and the mix of output by student level may well be implied by this present research. Although there are clearly limitations on the results which call for further research on the topic of costs in higher education, this thesis is the first to use economic analysis to explain and measure costs in institutions of higher education which has allowed for the possibilities of (d i s )economies of scale, interactions among the levels of student output and negative average variable and marginal costs using a weighted regression technique. It also shows that economic techniques and the assumption of rational behavior can yield useful results in yet another area of investigation. WORKS CITED WORKS CITED Fox, Karl A . , ed. Economic Analysis for Educational Planning. Baltimore and London: The Johns Hopkins University Press, 1972. Henderson, James H., and Quandt, Richard E. Theory. New York: McGraw Hill, 1958. Microeconomic Powell, John B. J r . , and Lamson,. Robert D. Elements Related to the Determination of Costs and Benefits of Graduate Education. Washington, D . C . : Council of Graduate Schools in the United States and National Association of College and University Business Officers, 1972. Tinbergen, Jan. Amsterdam: On the Theory of Economic Policy. North Holland Publishing Company, 1952. Unit Cost Study: Instruction and Departmental Research.,. 1962-6 3 , 1963-6 4 , 1964-6 5 . 1966-6 7 . 1968-6 9 , 1970- 7 1 . Lansing, Michigan: Michigan Council of State College Presidents, December 19— . Ziemer, Gordon, et.al. Cost Finding Principles and Procedures: Preliminary Field Review Edi t i o n , Technical Report Number 26. Boulder, Colorado: National Center for Higher Education Management Systems at W I C H E , November 1971.