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F 5.," .1“! -, :2',‘!'::,3 I”: ,...,., ,I.,I .- "Irma..- imam 2m :mx'mwnrcw....;3"2 ‘1‘”. . 1"..3: 4. H”.-. ‘I-f :37} 3...,“ _ 2-: u:-~ an...» ‘W LI-BRA {Y Michigan tam Univnis 'y \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 3 1293 10037 8425 mehmifiv-a:\u .4, .4 . x V This is to certify that the thesis entitled A MATRIX ALLOCATION PROCEDURE FOR UNIVERSITY COST ACCOUNTING presented by ALBERT A. EWALD has been accepted towards fulfillment of the requirements for Ph.D. degree in Business Administration fiémfliw [Ii/«cg gw/ Major profess’or/ Date NOV. 18! 1968 0-169 ABSTRACT A MATRIX ALLOCATION PROCEDURE FOR UNIVERSITY COST ACCOUNTING By Albert A. Ewald This study uses matrix algebra and random sampling procedures to estimate the costs to the University of South Carolina (USC) of providing to the students that education upon which the awarding of degrees is based. These costs are estimated for degrees by academic level for the eight Specific teaching (product) units and for the composite bachelor's and master's degree on a university—wide basis. In using matrix algebra this study devises a parti— tioned matrix containing four matrices] A, B, C and D. A allocates the outputs of nonteaching (service) units to other service units. B allocates the service units” outputs to the product units. C and D are the actual expenditures of the product and service units for the fiscal year 1967. Allocations are based on floor space, expenditures] student— load and/or opinions elicited from interviews. Allocations are not 100 per cent of a service unit's output where ser- vices are provided to other than the eight product units under study. Albert A. Ewald The partitioned matrix generates all the financial data used in this study. First the A matrix is inverted by the Gauss-Jordan procedure to become A_l. To check computa— tional validity, A is multiplied by A_1 to obtain the iden- tity matrix I. X, the product of A-lC, is multiplied by B to determine the value of service units' outputs transferred to product units——matrix F. D is added to F to obtain T, the total cost matrix of the product departments" outputs. Based on interviews, the percentage of each product unit's output devoted to teaching is obtained. This is multiplied by the total cost of each unit to obtain its teaching (net) cost. The credit hours delivered, by course and section number, are retrieved from the USC data bank. Based on interviews, each course of each college is weighted and a total number of equivalent units of output is obtained for each product unit. The net cost is divided by the equivalent unit out— put to obtain the equivalent unit cost for each product department. Multiplication of this by the weighting factor yields the cost per semester hour for each course. A random sample of lO per cent, or five transcripts if larger, was drawn from the population of each specific degree type awarded during the fiscal year 1967. The semes- ter hours completed for each graduate are classified accord- ing to semester hour cost. Accumulation multiplication of the number of hours by their cost equals the cost of each Albert A. Ewald degree in the sample. From this, the estimate of the aver- age cost of the specific degree and the estimate of the sampling error are calculated. Because of variations in sample size and sample means of degrees by type and level, stratified sampling techniques are applied to obtain an average cost of a bachelor”s degree and of a master's degree. Based on cost variations (from an input vieWpoint) administrators should consider whether future tuition charges should be based upon current practice or if they should be based on university average costs for the degrees, or on an estimated market value of the degrees to the recipients. Administrators of state—supported institutions should con- sider whether education should be provided to nonresidents below cost and what reciprocity arrangements should be made with other states. The inherent constraints of the university environ- ment impose limits on this work. The two major limitations are: capital charges for land rental and building deprecia- tion could not be obtained, therefore estimated degree costs are net of these charges; and, professorial time could not be isolated within the actual expenditures of the product units. As a result, all semester hours of credit produced by a college are charged with the same professorial cost although some variation exists with almost absolute certainty. A MATRIX ALLOCATION PROCEDURE FOR UNIVERSITY COST ACCOUNTING BY Albert A: Ewald A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Accounting and Financial Administration 1968 456/09/6/ \ 3~;/Z« ;; \ 5%? / ACKNOWLEDGMENTS Michigan State University through its Department of Accounting and Financial Administration provided me with an educational opportunity for which I am sincerely grateful. Dr. James Don Edwards, Department Chairman, provided encour- agement and support which exceeded any rightful expectation. The members of my dissertation committee, Doctors Gardner Jones (chairman), John Barson and Floyd Windal pro— vided a professional environment rich with constructive criticism and illuminating insights and imbued with a sense of scholarship. Their guidance can not be over-valued. The Department of Accounting and Financial Adminis— tration provided me with both financial support and the opportunity to work for Dr. Jones on the research project "A Procedural & Cost Analysis Study of Media in Instruc— tional Systems Development," of which this dissertation is an outgrowth. I am indebted to all these who supported me and especially to my immediate family and to the larger family of my professors and fellow students. I shall try, but this debt can not be paid in full. ii TABLE OF CONTENTS Chapter Page I. INTRODUCTION . . . . . . . . . . . . . . . . . 1 Purpose of This Study . . . . . . . . . . . 1 Need for This Study . . . . . . 2 An Outline of the Solution Method for the Cost Allocation Problem . . . . . . . . . 7 Scope, Methodology and General Limitations . . . . . . . . . . . . . . 9 Organization of the Thesis . . . . . . . . . 13 II. THE UNIVERSITY MATRIX SYSTEM . . . . . . . . . 15 Purpose of This Chapter . . . . . . . . . . 15 Background . . . . . . . . . . . . . . . . . 15 Definitions . . . . . . . . . . . . . . . 16 The Matrix System Analogue . . . . . . . . . 17 The USC Matrix System . . . . . . . . . . . 24 III. THE MATRIX SYSTEM CALCULATIONS . . . . . . . . 29 Purpose of This Chapter . . . . . . . . . . 29 Background . . . . . . . . . . . . . 30 The Addresses of Account Titles and Numbers . . . . . . . . . . . 3O Basis for Allocating Service Departments' Outputs . . . . . . . . . . . . . . . . . 3l Cross-Allocation of the Service Department"s Outputs . . . . . . . . . . . 38 Calculation of the Inverse . . . . . . . . . 39 C Partition Multiplication . . . . . . . . . 40 The Identity Matrix . . . . . . . . . . . . 41 T Matrix Calculation . . . . . . . . . . . . 42 IV. COSTS OF DEGREES . . . . . . . . . . . . . . . 44 Purpose . . . . . . . . . . . . . . 44 Product Departments Outputs . . . . . . . . 46 Equivalent Units of Output . . . . . . . . . 49 Cost per Equivalent Unit . . . . . . . . . . 50 iii Chapter Page Determination of Degree Composition . . . . 54 Degree Costs and Reliability . . . . . . . . 57 The Stratified Costs of Degrees . . . . . . 61 Statement of Results . . . . . . . . . . . . 64 V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS . . . 67 Purpose . . . . . . . . . . . . . . . . . . 67 Summary . .... . . . . . . . . . . .. . . 67 Conclusions . . . . . . . . . . . . . . . . 70 Recommendations APPENDICES . . . . . . . . . . . . . . . . . . . . . . 79 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 86 iv LIST OF TABLES Table Page 1. Addresses, account titles, numbers, and amounts of actual expenditures . . . . . . . . 32 2. Values of D, F and T matrices . . . . . . . . . 43 3. Percentage of total effort devoted to teaching . . . . . . . . . . . . . . . . . . . 48 4. Course weights . . . . . . . . . . . . . . . . 49 5. Equivalent units . . . . . . . . . . . . . . . 52 6. Semester hour cost matrix . . . . . . . . . . . 53 7. USC degrees awarded, fiscal year 1967 . . . . . 56 8. Cost of degrees . . . . . . . . . . . . . . . . 59 9. Table of statistical terms . . . . . . . . . . 63 10. Stratified costs of degrees . . . . . . . . . . 65 ll. Cost of degrees . . . . . . . . . . . . . . . . 7O LIST OF APPENDICES Appendix Page I. Inversion Program . . . . . . . . . . . . . . 79 II. Matrix Multiplier Program . . . . . . . . . . 83 III. Statistical Program . . . . . . . . . . . . . 84 CHAPTER I INTRODUCTION Purpose of This Study This dissertation describes an attempt to devise and apply a large cost allocation matrix in a university, as a means of systematically spreading indirect (service depart— ment) costs to producing (teaching) units whose primary out— put is degrees. The interdependency of dozens of service departments and the joint usage of common facilities make any more simple cost allocation procedure inappropriate. When costs allocated to a college or department by the matrix allocation are added to costs incurred directly by such a teaching unit, it becomes possible to identify the appropriate average unit cost of a degree produced by the teaching unit. The end—product of the costing technique attempted in this dissertation is a statement of the compar— ative costs of producing degrees from each of the several colleges within the university. The general purpose of this dissertation is to add empirical data to the information bank available to those who must solve the important problem of meeting the explod- ing demand for higher education with limited resources. Need for This Study The size and complexity of educational institutions have increased and continue to increase. The quantity and variety of degrees, research projects and publications--the products of these institutions--have also proliferated. This growth process involves elements both of horizontal and vertical expansion. Horizontal expansion is represented by the addition of a new college to a university. Vertical eXpansion is typified by the addition of a graduate degree program to an undergraduate degree program in the same aca— demic field. Background A modern university is comprised of many colleges which in turn are jointly supported by many service centers.* Services provided to the colleges by the service centers do not_account for the sum total of the costs of operating the service centers; some of their effort is used in serving each other. In fact, some service centers may exist to serve only service centers. In university accounting, ser- vice center costs directly allocated to the colleges——the ”product divisions”——are generally valued at some percentage of the budget of the contributing service center, if indeed *Service center is defined as a non—teaching unit whose services, nevertheless, are essential in the operation of a teaching institution. Examples are payroll, building maintenance, registrar. they are allocated at all. A more realistic valuation of a service center's aggregate contribution to the teaching units is its budget total less services rendered to other service centers, plus services received from other service centers. Three Basic Problems Three basic problems evolve from this situation. First, and most basic, is the problem of determining the total cost of any individual service center. The major obstacle in this problem is the large number of service centers involved in mutually supporting each other as well as teaching departments. Were but a handful of service cen- ters involved, the allocation of these cross—servicing costs would be mathematically easy and reasonably accurate. In actuality, however, there are in the order of 150 identifi- able cost centers in large universities1 of which several dozens may be service centers. Thus, an individual center may give and get services from each of the other dozens of service centers, and furnish services to the colleges. The traditional method of "successive allocation” in these kinds of situations is mathematically incorrect in that it ignores reciprocal services provided. The matrix allocation lU.S. Department of Health, Education, and Welfare Office of Education, A Procedural and Cost Analysis Study of Media in Instructional Systems Development, Part B, U.S. Department of Health, Education, and Welfare, 1964, p. 104. procedure by accommodating all the cross—services leads to a presumably more ”accurate" allocation.2 But even using a matrix procedure, the allocation of one center's services to up to 150 other centers and colleges requires exacting attention to causative relationships, which may, in fact, be quite elusive. Resulting allocations may be in error. By analogy, such an allocation is like cutting a pie into slices of one-hundredths of an inch with a knife that is thicker than the intended slice. The second basic problem is the disposition of the service center's cost once it has been allocated to the col— leges. Should the cost be allocated to undergraduate and graduate programs and to research? Should (can) graduate programs be divided into master's and doctoral levels for the purpose of charging them for allocated fixed costs? Should fixed costs be charged against research and publica— tions? In short, can the costs of products of a university be meaningfully differentiated? The third basic problem is that the expansion of programs and degrees has required increased and more varied services from the support departments ("service centers") of the institution. This increasing need has resulted in some service departments doing the bulk of their work for other service departments. Indeed, it is speculated that some Ibid. support departments may provide no direct service to degree— producing divisions but expend all their efforts in support— ing other service departments. (As a result, many contem— porary budgets for complex organizations require further analysis before their value as a planning and controlling device can be fully obtained. A budget for a department is in many respects an extrapolation of the expenditures of one or more prior periods modified for expected changes in activ— ities, student loads, or prices. All too frequently, how- ever, this modification does not take into full account the effect changes have on related support departments, espe— cially when the decision for change is initiated in the product department. The product department cannot usually foresee the "ripple" effects its decisions will have on ser— vice departments, which support it indirectly by servicing other service departments. This study may provide some in- sights about the effects of such decisions and may focus attention more clearly on the total cost of such decisions. Budgeting and Pricing Managerial attention is frequently applied to the differences between budgeted and actual expenditures. "Management by exception" draws attention to unfavorable variances, whereas favorable variances are frequently "avoided” by the spending unit's expenditure of all budgeted funds to ”protect” next year's budget. In some instances, unfavorable variances are avoided by having some effort performed by support departments. This study will tend to gather all costs and allocate them to the product depart— ments, thus drawing managerial attention toward total product department costs and their sources. Any manager likes favorable and dislikes unfavorable variances. But in a conglomerate organization such as a university there may be product lines (degree programs) with seemingly favorable operating characteristics because the allocation procedure for costs favors some products to the disadvantage of others. Product lines (such as fields of study) are being added to such organizations because of their social worth or "image-building" value. What is needed is a more pragmatic and empirical basis upon which to budget or commit the usages of assets. This proposed research will make some contribution toward this end. The main product line of a university is degrees composed of credit hours of instruction. Different degrees have different compositions of credit hours. Credit hours in some academic fields are expected to have different costs than hours in other fields. This study will attempt to generalize about the costs of various ”kinds” of credit hours and to summarize the total cost of providing the instruction for different degrees. It is anticipated that, for example, a degree in an exotic and highly Specialized science will cost more than a degree in a routine and highly stylized field. This study may provide a pragmatic and empirical basis upon which university administrators can budget and commit assets to such usages. It is most probable that the values of degrees, from a human capital point of View, vary widely. The value of one degree may have a much higher market value to its recip- ient than another degree to its recipient. This study may provide a point of departure for further studies into the pricing policy for various degrees and for more quantitative comparisons of the actual costs of programs with their social worth. Ultimately, help for answers to these inter- related questions may emerge; for example, should the price of a degree be based on its value to the recipient? Should the price be based on cost? Should the costs be borne by the larger community because of the social benefits derived from such programs? An Outline of the Solution Method for the Cost Allocation Problem The problem studied in this work will be stated in a general way and then in more detail. The specific methodol— ogy will be given in the next section. In general, the problem solution is first to allo— cate the indirect costs of a university to its product- producing divisions. Second, these allocated costs are added to direct costs of the producing divisions to arrive at a total cost for all products. Third, these products are classified into categories to enable costs to be assigned to them. Fourth, the cost of the product——be it a bachelor's, master's, or doctor's degree—-can be compared to its price, and pricing and other implications can be drawn. As more and more such studies of degree costs are undertaken, re- sults of this and future researches may indicate a measure of central tendency for the costs of each of the various products of a university. In the first step in the solution of this problem, the recipients of the outputs of service departments are divided into two classes-—service departments and product departments. Second, the output of each of the service departments is assigned to the recipients on the basis of the most closely identifiable usage relationship. In the third step the matrix of simultaneous equations is con— structed that states the allocation pattern; this system is then solved for its inverse. Once the inverse is obtained, the total costs of the product departments are obtained by a straightforward matrix multiplication. The products of the producing departments are identified and measured. They fall into the categories of delivered credit hours of graduate and undergraduate instruction, research, publications and other such scholarly endeavor. For each department, its total product costs are assigned to the several products, including the several degree categories. The differences between the costs of different degrees are calculated. The difference between the costs of and prices paid for various degrees are exam- ined, conclusions are drawn and recommendations are made. Scope, Methodology and General Limitations Scope This study will be limited to a one—year (fiscal 1967) operating cycle of the University of South Carolina. Only those of its accounting records which are in the public domain will be used. Consideration of the university's products will be limited to the undergraduate and graduate degrees. These degrees will be classified and examined on the basis of the college which awards them and by their academic level, but no differentiation will be made among various degrees at the same level from the same college. The scope of this work, therefore, will be limited to those colleges which provide accredited instruction. Colleges which provide non—accredited training, if dealt with at all, will be treated in an incidental way for purposes such as clarity or possible future studies. A study of one school for one year may seem to impose debilitating geographic, curriculum and time limita- tions. This is not the case. Professors, equipment, sup— plies and other factors used by a university for producing instruction are taken from regional, national and interna- tional markets. Curricula for similar degrees in different schools are acceptably similar in composition. In this 10 study, the matrix model is so constructed that the budget data for any year can be inserted to find current or future degree costs. This study, because it determines one cost for one "composite" degree at each academic level from each college, will result in the same cost for very different degrees. This imposes a limitation. For example, the College of Arts and Science awards approximately twenty various degrees at the undergraduate level. Thus a degree in English or Chemistry or Social Science or Physics will have the same "composite" degree cost. The probability that all these degrees would in fact have the same cost is not high. There— fore, this study will generate cost data which are valid for "composite" degrees rather than cost data for specific degrees. Methodology The methodologies to be employed in this research fall into three clusters: the cost allocation techniques, the computer techniques and the statistical sampling tech- niques. Each of these techniques has some associated limitations. The allocation of the outputs of the service depart— ments will be based on interviews with the most knowledgeable personal available, the floor space occupied, the student load and/or the amount of the department's expenditures. The allocation of the efforts of the product departments 11 will be based on interviews as will be the weighting of the teaching effort for various levels and types of instruction. Such allocations are tenuous and dependent on memory. They are, however, the best available. The computer techniques to be used in this study are the Gauss—Jordan Matrix Inversion Procedure and several pro- grams that multiply matrices and retrieve data from the cen- tral data bank. Computer usage introduces limitations. The computer truncates rather than rounds numbers thus introduc— ing some minor errors. The matrix inversion and multipli- cation programs cannot be verified, practically, by hand. This limitation will be attenuated by solving a similar problem, whose answer is known, with the programs employed. The matrix inversion technique will be tested further for validity by a matrix multiplication that should produce the Identity Matrix. This should overcome the technical limita- tions of computer usage. However, there is no practical way to verify the tapes in the central data bank of the univer— sity and this study relies on them for the number of semes— ter hours of credit produced by the colleges. They are assumed to be reliable. This research will employ a stratified sampling technique as a basis for constructing the statistics rele— vant to the costs of the various degrees. The sample size for each "composite" degree will be 10 per cent or five samples, whichever is larger, without replacement of the 12 number of degrees, by academic level, granted by each col— lege and school. Samples will be randomly selected. Since the stratified sample will not be exactly 10 per cent for each degree, the larger than 10 per cent samples will have diSproportionately more weight. This will skew the cost slightly toward the less popular degrees. General Limitations This research requires identification of the prod— ucts of a university and the costs which become embodied in these outputs. Such research also requires that the efforts which go into a particular output be sorted out and allo- cated to that product. If the product is a bachelor's degree, for example, it is nearly impossible to segregate specifically the degree yielding efforts from those yielding research results. Yet, some effort to do so is necessary regardless of the inherent and unavoidable inaccuracy of the results. The effort will at least provide a point of depar— ture toward greater future accuracy. The problem of sorting out costs requires the split— ting of joint costs and measurement of intangible, invisible joint products. Further, the university is the type of orga— nization where personnel and accounting structures are not geared to cost accounting and where almost every resource is used jointly. This environment is one in which accounting systems and procedures are designed for assuring budgetary compliance and in which cost accounting is quite foreign to 13 employees, especially faculty. To secure the information necessary for costing is of major importance to the problem because if the purpose is misunderstood the effort will be vigorously resisted. Top-level administrative support is essential. The first step toward a definitive product—costing system for a university is the allocation of the support activities which the non—teaching organizational units (indirect costs) provide to the organizational teaching units. No really satisfactory basis of allocation exists but there are two reasonably satisfactory approaches. The first is the traditional service center allocation approach which is efficient when there are a few cost centers and the second is the matrix allocation procedure which is more suit— able to large allocation problems. This work will use the matrix allocation procedure for indirect costs to determine a basis for recognizing cost differences in different sub— ject matter areas. Some pricing implications should follow. Organization of the Thesis This research comprises four additional chapters each of which begins with a section that explains the pur- pose and nature of the chapter. The next chapter, the second, devises and defines the system of matrices and a matrix address system. In the third chapter, the matrix system is identified and the calculations required to 14 compute total costs of the product divisions of USC are performed and tested for accuracy. The fourth chapter will determine the average costs of each of the types of under- graduate and graduate level degrees awarded by USC and the average cost of the average or composite undergraduate and graduate degree. The last chapter, the fifth, contains the summary, conclusions and recommendations. The computer programs employed in this study will be placed in the appendices in the order of their appearance. CHAPTER II THE UNIVERSITY MATRIX SYSTEM Purpose of This Chapter The purpose of this Chapter is to devise and define a system of matrices into which selected accounts of the University of South Carolina will be arranged for matrix calculations. This system will include an input set and an output set of matrices. The input set will be arranged into a partitioned matrix which will generate the set of output matrices. A further purpose is to design the address system by which a matrix element can be efficiently and precisely located in the partitioned matrix space. ,In this chapter some inference will be made to some of the allocation values and actual expenditures data which are given in Chapter III. Background Matrix algebra is covered comprehensively in many standard textbooks and is not, per se, within present scope. From those many standard references this work relies 15 16 primarily on four. The first by Kemeny et a1.1 provides good coverage of vectors and matrices. Chris Theodore2 pro— vides similar treatment. Howell and Teichroew3 provide a straightforward and clear treatment of matrix algebra which is reproduced by Mattessich.4 This study follows the con— ventions of matrix algebra outlined in these and the other standard textbooks dealing with the subject. Definitions The matrices used in this and the following chapters are verbally and, where relevant, mathematically defined as: A — the allocation of the service departments‘ outputs to other service departments A_l — the inverse of A B - the allocation of the service departments' outputs to the product departments C — actual (or budgeted) service department expenditures 1 John G. Kemeny, Arthur Schleifer, Jr., J. Laurie Snell, Gerald L. Thompson, Chapter V "Vectors and Matrices,” Finite Mathematics with Business Applications (Englewood Cliffs, N.J.: Prentice Hall, Inc., 1964), pp. 229—311. 2Chris A. Theodore, Chapter 9 I'The Algebra of Linear Programming," Applied Mathematics: An Introduction (Home- wood, Illinois: Richard D. Irwin, Inc.), pp. 332—369. 3James E. Howell and Daniel Teichroew, Mathematical Analysis for Business Decisions (Homewood, Illinois: Richard D. Irwin, Inc., 1963), pp. 234-246. 4Richard Mattessich, Accounting and.Analytical Methods (Homewood, Illinois: Richard D. Irwin, Inc., 1964), pp. 466—478. 17 D - actual (or budgeted) product department expenditures I - the identity matrix F — the value of the outputs transferred from the service departments to the product departments, BX = F P - the partitioned matrix containing A, B, C, and D. The input set T - the total cost of the product departments' outputs. T = D + F X — the unknowns CA"l = X* In this work the matrix elements are identified by the lower case of that letter which identifies the matrix to which they belong. For example, aij is an element of A, bij is an element of B, etc. In this work, actual expenditures are used. They can be replaced readily by budgeted expenditures when actual values are not available, or when administrators wish to use estimated data to determine their behavioral effect on costs. The Matrix System Analogue The Analogue The large size of the matrix system of USC impairs clear discussion and manual algebraic operations. Its the unknowns is derived from: X I AX = C, thus X = C/A. But the matrix equivalent to the reciprocal of A (the inverse) is A' , hus x = CA-E. 18 system, however, can be simulated by a small analogue whose matrices are consistent with the definitions and notations given above. Assume a situation in which three service depart— ments cross-service each other and provide services to two product departments. Each service department provides 20 per cent of its output to each of the other service and product departments. Every department has $100 of actual (or budgeted) expenditures. The Matrix Arrangement The first objective of the matrix arrangement is to gather into the A, X and C matrices those accounts having a reciprocal relationship. Reciprocity is the mutual giving and getting of support among service departments. These equations state this relationship in the analogue: x1 = 100 + 0.25x2 + 0.25x3 x2 = 0.25xl + 100 + 0.25x3 (2—1.0) x3 = 0.25xl + 0.25x2 + 100 in which x1, x2 and x3 are service department costs. Equa- tion system (2-1.0) can be rewritten as: 1.OOXl - 0.25X2 - 0.25X3 = 100 -O.25xl + l.OOX2 - 0.25X3 = 100 (2—1.1) -O.25X - 0.25X + 1.00X = 100 1 2 3 19 This converts into the following matrices: _ _ _ I F _ 1.00 —0.25 -0.25 x1 100 -0.25 1.00 —0.25 x2 = 100 (2-1.2) -0.25 —0.25 l.00__& x3__ _100J or in matrix notation: AX = C (2—1.2M) The second objective is to gather into the B, X and D matrices the accounts which have a non-reciprocal relation— ship. These accounts, the product departments, are non- reciprocal because they get support from but do not give support to the service departments. These equations state this relationship in the analogue: (1- II 100 + 0.25x + 0.25x + 0.25x 1 2 3 + 0.25x2 + 0.25x (2—2.0) t = 100 + 0.25x 1 3 in which t1 and t2 are the costs of the respective product departments' outputs. These equations can be manipulated to get: tl — 100 f1 t — 100 f x 0.25x + 0.25x + 0.25x 1 2 3 l 0.25xl + 0.25x + 0.25x (2-2.]J 2 3 in which fl and f2 are the values of the outputs transferred from the service departments to the product departments. Equation system (2-2.1) can be rewritten as: 20 , _ t a f 0.25 0.25 0.25 x 1 _ 1 = 1 l (2-2.2) t2 a2 f2 0.25 0.25 0.25 x2 or in matrix notation as: T - D = F = BX (2—2.2M) This equation states that the total cost of all products less the actual (budgeted) expenditures of the product departments equals the value of the outputs transferred from the service departments. The Address System The partitioned matrix P used in this dissertation, when adapted to the analogue is: ,— a a | C a11 12 13 : 14 i a21 a22 a23 5 C24 I a31 a32 a33 : C34 I l i b41 b42 b43 : 44 I _?51 b52= b53 : d54 In this format, the section containing the a's constitutes the A matrix, the sections of the b's,_c's and d's consti- tute the B, C and D matrices. The subscripts identify the row and column positions (the addresses) of the a, b, c and d elements in the partitioned matrix. 21 In equations (2-l.2) and(2—2.2) A, C, B and D are the input matrices of the partitioned matrix P. In the A matrix the conventional matrix addressing system is aij’ (i = 1,2,3; j = 1,2,3) where the i value locates the row and the j value locates the column position of the specific matrix element. In this work this convention will be fol— lowed for the A matrix. The C matrix normally would have the conventional address system Cij’ (i = 1,2,3; j = 1). In this work its address system is modified so that cij’ (i = 1,2,3; j = 4). Thus, the j value of this one column vector is one number more than the highest j value in the A matrix. In the B matrix the conventional address system is b.. (i = 1,2; j = 1,2,3). In this work the address system 13 is b.. (i = 4,5; j = 1,2,3). Thus, the 1 row of the B 13 matrix begins with a number one higher than the highest 1 value in the A matrix. Similarly, the D address system becomes dij (i = 4,5; j = 4). Thus, any D matrix begins with an i and j value one higher than the highest i and j, respectively, in the A matrix. This system locates the articulated set of matrices and their elements in the par- titioned matrix P. The articulation of the matrices becomes clearer when the matrix space is duscussed. 22 The Partitioned Matrix Space 'In the analogue the A matrix is square and comprised of three row and three column vectors. Had the analogue con- sisted of n service departments the A matrix would have been square and comprised of n row and n column vectors. It is \ necessary to maintain squareness to invert the matrix. Squareness is maintained by assigning a coefficient of 0 to service departments from whom or to whom cross-servicing is not provided. For example, in the analogue, if department al received no support from department a2 but instead depart- ment a received that incremental 25 per cent, the A matrix 3 of equation (2—l.2) would have been: 1.00 0 --0.25 A = —0.25 1.00 -0.25 (a-1.2a) -O.25 -O.50 1.00 The matrices of the partitioned matrix articulate with each other in several ways. Each column in P, except the last, represents the total allocation of a specific ser— vice department. In each allocating column, the net value of the matrix elements in the A partition equals the value of that service department's output which is available to the product departments. The allocation of this output is expressed by the elements in the same column of the B parti- tion. If the system is exhaustive, all this output available 23 in a column is allocated to the receiving product depart— ments in the B matrix partition of that column. The A, B, C and D matrices are articulated in this section identically as are the much larger matrices of the USC problem. The arrangement of the analogue matrices is: —— —————————————————————————————————— dfi A C .., 1 = l, 2, 3 .., 1 = l, 2, 3 ij 13 j = 1, 2, 3 j = 4 __ .1 equation (2—l.2) equation (2-l.2) P j = 1, 2, 3 Substitution of the values from the matrices of the analogue into their address in the partitioned matrix yields: 3 = 1 2 3 4 F’ ' ’l 1 1.00 -0.25 —0.25 I 100 l 2 -0.25 1.00 —0.25 I 100 l 1 = 3 —0 25 -0.25 1.00 l 100 ________________________ +___.___ 4 0 25 0.25 0.25 l 100 l 5 0.25 0.25 0.25 l 100 _. l .— 24 The USC Matrix System USC is divided, for the purpose of this work, into two subsets. The producing subset consists of 10 elements which are located on campus and 23 elements which are widely dispersed throughout the state. The remainder of the Univer— sity's activities comprise the subset of service departments. The 10 On-campus elements include 8 colleges and schools which produce college-level instruction, the College of General Studies which produces noncollege level instruction and the Department of Advanced Studies and Research which does not produce instruction. The 23 off—campus are 11 re— gional centers, 9 hospital affiliates and 3 field activities. Of these 33 elements, only the 8 colleges and schools which produce college-level instruction are relevant to this study because only they award degrees. The 25 remaining elements will be combined in one category for the purpose of assign- ing to them the service departments' outputs which they absorb. This will be accomplished by the allocation process of Chapter III in which only services absorbed by the 8 rele— vant elements will be carried forward. The A Matrix Schema There are 42 activities in the service department subset. Each service department has a number which sum- marizes subsidiary departmental accounts, in the chart of accounts. 25 These 42 activities, mutually supporting each other, constitute the A matrix that is 42 x 42 in dimension. The account numbers and titles of these activities will be iden- tified by their matrix location number in Chapter III. The schema of the matrix is such that for any aij’ i = j. Thus, for example, a service department, say Personnel Services, with an address which is aij = pij’ (i = j = 5), is the fifth row vector of the matrix for allocation of services it receives and the fifth column vector for allocation of services it gives. This ordered arrangement ensures the required squareness of the A matrix and permits easier manipulation with an identity matrix. The B Matrix Schema There are 8 members of the product division subset that receive support from the service subset. These product divisions are arranged in an 8 x 42 matrix, the B matrix, and will be identified in Chapter III in the same way as the product departments. The schema of this matrix is such that for a bij = pij (i = 43, 44 ..... 50; j = 1,2, ..... 42). The address 1 locates the row vector which receives and j locates the column vector which provides services. Thus bij = pij (i = 44, j = 5) locates the percentage of Personnel Services' output received by some product department, say the Business College. 26 The C Matrix The C matrix is a single column vector of 42 rows. Each of the 42 elements is the actual expenditure of one of the service departments. The values were taken directly from the accounts of USC. The format of this matrix is cij = pij and i = l, ..... 42 and j = 43. Thus c5,43 = p5,43 locates the amount actually spent by the Personnel Services Department as will be shown in Chapter III. The D Matrix The D matrix is, like the C matrix, a single column vector but it has 8 row elements. Each row element is the actual expenditure of a college or school as taken from the USC accounts. The format is dij = pij and i = 43, ..... 50 and j = 43. Thus d44,43 = p44,43 locates the amount actually spent by the Business College as will be shown in Chapter III. The Partitioned Matrix Space The row dimensions of the P matrix is equal to the sum of the product and service departments or 50. This is equal to the highest i value in the system. The column dimension is equal to one more than the number of service departments or 43. This equals the highest j value in the system. The general schema of the P matrix composed of the matrices--A,B,C and D——of the main allocation problem is: 27 j = l, 2, 3 ...................... 42 43 1 = 1 A = .. = b C = 13 l] 2 = ........ .. = 1 1, 42 13 pl] 3 j = 1, ........ 42 1 = 1, ..... 42 j = 43 42 43 B = bij = pij D = dij = pij . i = 43, ......... 50 i = 43,....50 . j = 1, .......... 42 j = 43 50 The matrix will differ from the partitioned matrix of the analogue in one way. The allocations made in the analogue are exhaustive. As a result, the algebraic value of the portion of any column falling within the A section of that partition will equal the algebraic value of that same column falling within the B section of that partition. In the USC matrix the allocations are not exhaustive. The services absorbed by the 25 ”other" product departments, although carefully calculated are not carried forward because they are not relevant to this work. For these departments 28 their combined proportional absorption of services causes and is equal to the algebraic difference between the two parts of any such column in the USC matrix. CHAPTER III THE MATRIX SYSTEM CALCULATIONS Pugpose of This Chapter The overall purpose of this chapter is to compute the total costs of the product divisions of USC. This will be accomplished in five steps. First this chapter will identify the account titles, numbers, and actual expendi— tures of the product and service departments of USC with their address in the partitioned Matrix P. Next, the allo- cation bases for the service departments' outputs to the service departments and product departments will be dis- cussed. This second step will include allocation of service departments' outputs to the recipient departments. This will complete the P—Matrix. The third step will be to determine A_l, the inverse of the A partition of the USC Matrix. This will be accomplished by using the Gauss—Jordan Procedure. Also in this step the C partition will be multi- plied by the inverse to determine X, the Matrix of unknowns needed for calculating F, the matrix of fixed cost. Fourth, a test of these calculations will be made in which A will be multiplied by A.1 to get the Identity Matrix, thereby to prove the quantitative integrity of the computer calculations. 29 30 The fifth and final step will be to add the F and D matrices to obtain the Total-Cost Matrix, T. Background The work in this and subsequent chapters will use the symbols as defined on pages 16 and 17 of Chapter II above with no further definition. New symbols and defini- tions will be added as needed. The Addresses of Account Titles and Numbers The P Matrix has 50 row locations of which 42 are in the A partition. This square partition, A, is arranged, as eXplained in Chapter II, so that the row and column loca— tions of service departments are identical. Thus, the 42 service departments of USC, which comprise the A partition, are so arranged that each row or 1 location of an account is the same as its columnar or j location. As a result, the address 1 = j obtains for each service department. The remaining 8 row locations of the matrix contain the product departments. The product departments are iden- tified by row location only. As a result, the address i for each product department has no counterpart j address. The 8 x 42 quadrant containing the product departments is the B partition. This and the A partition completely define the 50 row locations or addresses. 31 Of the 43 column addresses or locations, 42 have been defined as elements of the A partition. The 43rd column, a 50 x 1 vector, makes up the C and D partitions. This column vector articulates with the row identities in the following manner. Column 43 contains the actual expen- ditures in its appropriate ith row for the ith account num— ber and title. Thus the address pl,43 locates the first row account title and number and column 43 of that row states the amount of actual expenditure of that account. In this work, the addresses, account titles, numbers, and amounts of actual eXpenditures are given in Table 1. Basis for Allocating Service Departments' Outputs The allocation of each of the outputs of the 42 service departments of USC to the other 41 service and to the 8 product departments involves 42 x 49 or 2,058 decisions. To discuss succinctly such an enormous allocation process requires a systematic treatment. An eclectic approach was taken in allocating the outputs of the service departments. At the outset interviews with various administrators were used to determine if some empirically based method of allocation existed. The empir- ical methods for allocation were based on occupied floor space, actual direct departmental expenditures and number of credit hours of instruction completed. In some cases there was no sensible empirical basis for allocating either a part 32 Table 1. Addresses, account titles, numbers, and amounts of actual expenditures Address Account Actual i j Number Account Title Expenditures 1 1 01000 Board of Trustees 55842.00 2 2 01100 President 63180.00 3 3 01150 Senior Vice-President 32831.00 4 4 01200 Vice Pres. for Bus. Affairs 70982.00 5 5 01201 Personnel Services 30920.00 6 6 01202 Accounting Services 133905.00 7 7 01205 Purchasing & Central Supply 104136.00 8 8 01207 Auxiliary Services (Admin.) 6000.00 9 9 02001 Vice Pres. for Student Affairs 102624.00 10 10 02002 Student Union 58303.00 11 11 02003 Counseling Bureau 5204.00 12 12 02004 Student Aid 20307.00 13 13 02006 Admissions & Registration 240571.00 14 14 02100 Vice-Pres. for Development 29541.00 15 15 02101 Alumni Association 9171.00 l6 16 02102 Placement 14657.00 17 17 02150 Public Rel. & Information 36532.00 18 18 02201 Mail & Messenger Service 40067.00 19 19 02202 Printing 113306.00 20 20 02203 Telephone Exchange 3988.00 21 21 02204 Data Processing Service (Non-Acad.) 156860.00 22 22 02209 General (Association Dues) 4313.00 23 23 03000 General Instruction 84345.00 24 24 03020 Associate Vice—President 30955.00 25 25 03030 E.T.V. Center 28064.00 26 26 03030 Computer Science Center 210769.00 27 27 03040 University Press 68717.00 28 28 03100 Vice-Pres. for Academic Affairs 82944.00 29 29 07000 Administration & Supervision 68068.00 30 30 07001 Maintenance of Buildings 730865.00 31 31 07002 Maintenance of Grounds 144158.00 32 32 07003 Custodial Services 218910.00 33 33 07004 Police 96277.00 34 34 07006 Heating Plant 103966.00 35 35 07007 Utilities 110103.00 36 36 07009 Motor Pool —9835.00* 37 37 07012 Insurance (Admin. & Acad. Bldgs.) 20079.00 38 38 07020 Campus Planning 54507.00 39 39 08400 Health Service 103205.00 40 40 06000 Libraries 668582.00 41 41 08100 Director of Housing 1654252.00 42 42 08500 Cafeterias 68016.00 43 43 03200 College of Arts & Science 3967256.50 44 43 03300 College of Business Admin. 751477.19 45 43 03400 School of Education 611504.46 46 43 03500 College of Engineering 561063.29 47 43 03700 School of Journalism 122593.38 48 43 03750 Law School 197989.12 49 43 03800 School of Nursing 225861.11 50 43 03850 School of Pharmacy 127543.81 *The Motor Pool collects charges from (i.e., transfers its costs to) users of its services. The rates set for its services during the fiscal year 1967 were too high. As a result, the motor pool account accrued credits in excess of its expenditures, or in other words, it transferred more than its actual costs to users of its services. 33 or the entire output of some departments. In these cases, the administrators were asked to allocate such departmental outputs based on their best judgment. Some further distinction needs to be made between the general process of interviewing and interviewing as defined here is a method for allocation. In this disserta— tion, the interview method of allocating means that the actual allocations are based on the opinions of the inter- viewees. It is not an empirically based method and is therefore subject to more chance of error; there are no tangible, measurable, arithmetic bases beyond the proper— tions indicated by opinion. Interview Method A11 interviews had one underlying guideline. Inter- views were sought with administrators at a high enough level for them to be knowledgeable of the subject matter but not at a level so high that the subject was trivial to them. The highest administrative officer interviewed was the vice- president for business affairs; the highest academic officer was the vice-president for academic affairs. As a result of these interviews certain allocations were made on the basis of the interview method. Such alloca— tions were made for either all or a portion of the output of these departments whose partitioned matrix column addresses are j = 1, 2, 3, 4, 5, 7, 8, 10, ll, 12, 13, 14, 17, 18, 20, 21, 24, 25, 26, 28, 29, 38 and 39. 34 If a department had its output allocated wholly on the basis of the interview method its column address will appear only in the preceding sentence. If its output is allocated partially on the interview method its column address will reappear in one or more of the empirical list- ings given below. A department partially based on the inter- view method occurs when some portion of its output is devoted to another department—-in the opinion of the interviewee——and the balance of its output is allocable on some empirical basis. Floor Space Occupied The interviews with the most knowledgeable adminis- trators revealed that some service departments should have their output allocated on the basis of floor space. For this allocation, the latest available blueprints of the USC buildings show that 2,730,140 square feet of Space were being used. The floor space occupied by each department was converted to a percentage of the total USC Space. However, during the year much construction was in progress and in the future some departments will have their locations changed both in locus and size. The allocation is reasonably valid for the period of this study but future work Should incorpo- rate new construction into the floor Space base and reassess the space occupied by the departments. 35 Use of floor space occupied as a basis for alloca- tion carries with it the associated problem of allocating maintenance, repair and utilities eXpenses. Newer buildings usually will have lesser maintenance eXpended on them than older buildings and will use lesser amounts of utilities because of more efficient insulation, lighting, building design, etc. The capital costs of newer buildings, however, will offset the occupancy costs because the unamortized in- vestment in the new usually will be larger than the residual investments in the old buildings.1 Specific identification of all these costs of the individual buildings is of impor- tance to administrators who must decide on the design of new structures. For the purpose of this work, however, the ”accident” of a department's location is irrelevant to the main purpose, computing the cost of delivering instruction. The results are more generally valid to present purpose if all floor space is treated as if it were homogeneous both in terms of capital investment costs and occupancy costs. These departments with newer facilities have their lower occupancy costs offset by higher capital investment costs: the higher occupancy costs of old facilities are offset by their lower capital investment costs. In this manner, the ”accident” of physical location and the effects this has on 1In other Situations capital costs might be based on amortized costs of depreciable assets. 36 a department's cost tends to be mitigated and the overall results become more generally valid. Departments whose outputs are allocable, in whole or in part, on the basis of floor space have the column addresses j = 30, 31, 32, 33, 34, 35 and 37 in the P matrix. Credit Hours Some departments provide services which are propor— tionate, in whole or in part, to the number of hours of instruction delivered. For a diversified university, such as USC, hours of instruction fall into three categories. First, some outputs are proportional to the total instruc- tion delivered throughout the university system. This in— _cludes the instruction delivered in the centers, regions, and on campus. The second category comprises those outputs proportionate to the number of hours of instruction delivered on campus whether for regular college credit or not. The third category comprises on—campus instruction which is accredited college work. This third category is the actual product for which this work strives to determine a cost. Examples of an output allocable on the basis of total hours taught in the university system are the Counsel— ing Service and the President's office. Purchasing is pro- portionate to semester hours on campus; only the regions and centers are self supporting in this function. The Alumni Association is an example of a service proportionate only to on-campus credit hours of instruction. Allocations based on 37 total hours of instruction, in whole or in part, are appli- cable for departments whose columnar addresses in the P matrix are j = 2, 11, 13 and 17. Allocations based, in whole or in part, on total hours of instruction on campus are made for addresses j = 5, 7, 9, 10, 12, 16, 18, 20, 22, 23, 27, 28, 39, 40, 41 and 42. The total hours of on campus college credit instruc- tion was the basis for allocating, in whole or in part, the output of only the Alumni Association whose address is j = 15. Actual Expenditures or Budget Some activities of the service departments of USC are proportionate to the amount of expenditures made by the recipient of these services. For example, the USC controller reported that his department rendered varying amounts of its output to differing groups of recipients depending on the time of year, the Specific year in question (and construc- tion activities currently being carried on). His considered opinion is that for general applicability and long range validity his services should be charged on the basis of the amount of money spent by the departments being serviced. Departments whose services are allocable on the basis of eXpenditures have the addresses 1 = j = 6, 19 and 36. 38 Cross-Allocation of the Service Department's Outputs Thus far this chapter has established the address system for the partitioned USC matrix and identified the 42 service departments by account number and title. The amounts of departmental expenditures have also been given in Table 1. The various methods for allocation have been given in a general way. AS a result, all that remains to cross allocate service departments' outputs is to complete the A partition of the USC matrix. The actual allocations in the A partition entail 42 x 41 or 1,722 decisions. Each decision becomes one ele- ment in the 42 x 42 partition. These allocations are not listed in this dissertation because they are not generally applicable. They are relevant only to USC for the time period under study. The allocation values are analogous to the values listed in the A partition of the matrix of the analogue on page 23 of Chapter II above. The equations upon which the allocation values are based are linear. There are 42 of these equations and 42 unknowns. These equations are of the general form X1 = C1,1 + a1,2X2 + °°°°°°° + a1,42 X42 2 (3-1.0) X = a + a x + ........ + C 42 42,1 42,2 39 In this system, Cij represents the actual departmental expenditures of the ith service department in the one column C partition of the USC matrix; aij represents the aij = pij coefficient in the 42 x 42 A partition of the USC matrix;2 and, Xi represents the ith unknown in the one column X matrix of unknowns as defined in Chapter II. Following the same algebraic steps as used in the Analogue, these 42 equa- tions become, in matrix notation: A X = C (2—1.2M) The system of equations in 3-1.0 does not exhaust the entire output of the service departments. The remainder is allocated among the product departments in Chapter V below. Calculation of the Inverse This dissertation capitalizes on the work of Williams and Griffin,3 particularly on their application of the matrix inversion procedure for cost allocation. Their matrix is 5 x 5, which is much smaller than the 42 x 42 matrix in the A partition of the USC Matrix. Their smaller matrix was used in test runs of the computer programs used in this work. 2Because of algebraic operations, however, these coefficients will be of different Sign. For example, see equation 2—l.2a. 3Thomas H. Williams and Charles H. Griffin, "Matrix Theory and Cost Allocations," The Accounting Review, XXXIX, No. 3 (July, 1964), 671-678. 40 The results obtained were consistent with those found by Williams and Griffin. Inversion of a matrix, when a computer is used, iS procedurally routine. After the allocation values have been determined, as described above, the balance of the work is voluminous but straightforward. It consists of key punching data cards, writing the computer program, running the data through the computer and proofreading the printout of the input data. The Gauss-Jordan procedure, a standard computer library program, is used to invert the A Matrix. This inver— sion program is given in Appendix 1. The inverse, A-l, is an intermediate result. Because its main purpose is to serve as a matrix multiplier of the C partition, the A_1 Matrix is not reproduced in this work. The result of the A"1 multipli- cation of the C partition becomes the X matrix, dealt with below. C Partition Multiplication The multiplication of the C partition by the inverse is a straightforward calculation, but a very large one. The multiplication by the 42 x 42 inverse matrix of the 42 x 1 C matrix entails 1,764 multiplications and the addition of 42 columns of figures, each containing 42 values.4 Part of —1 4This multiplication in matrix notation is equivalent to A C = X. 41 the value of computer applications is to make it unnecessary to do such laborious work manually. The program listed in Appendix I includes instructions for the computer to perform these operations. The result of these calculations is the X Matrix, which is multiplied by the B partition to determine the "actual allocation” of the service departments' outputs to each of the product departments. The Identity Matrix The calculations performed by the computer to invert the A partition if done manually would require at least several man-years of effort. Because of this it is not feasible to check the computations manually, nor is it satisfying to accept blindly the computer's results. It is possible to authenticate the results by observing that a matrix, if multiplied by its inverse (its matrix-algebra reciprocal), yields the identity matrix. As a brief proof of the matrix multiplication pro— gram of this work, this program was used to multiply the 5 x 5 matrix of Williams and Griffin by its inverse and it yielded the identity matrix.5 5T. W. Williams and C. H. Griffin, op. cit., p. 676. 42 The A partition of the USC Matrix when similarly multiplied by its inverse does not yield the exact identity matrix. In several elements of the approximate identity matrix there are values that appear larger in the fifth place to the right of the decimal point than the exactly accurate 0.00000 element value. This acceptably small error results primarily from two causes. First, the allocation of the service departments' outputs were rounded to the fifth decimal place. Second, the computer truncates rather than rounds values. The program used for multiplying the matrix by its inverse is given in Appendix I. The results of the multi— plication are deemed to validate the calculations. Thus far in this chapter the A, B, C and D matrices have been defined, and the A-1 and X matrices have been cal— culated. The work remaining to be accomplished is to compute the F Matrix and add it to the D matrix to obtain the T Matrix. T Matrix Calculation The F Matrix is obtained by multiplying the X Matrix by the B Matrix as outlined in Chapter II above, with Spe— cific reference to equations 2-2.2 and 2—2.2M. The actual calculation involves 42 x 8 =336, multiplications and addi— tions which are done by the computer to capitalize on its high speed. The program for this multiplication is, except 43 for dimensions, Similar to the program portion which multi— plies A-1 by C. This program is given in Appendix II. The F Matrix when added to the D Matrix yields the T Matrix. These values are given in Table 2 below. Table 2. Values of D, F and T matrices Product Department Address D F T i j 43 43 $3,967,256 $3,516,629 $7,483,885 44 43 751,477 831,124 1,582,601 45 43 611,504 390,101 1,001,605 46 43 561,063 239,914 800,977 47 43 122,593 52,154 174,747 48 43 197,989 163,030 361,019 49 43 225,861 85,152 311,018 50 43 127,543 94,175 221,718 The valuescfifthe T Matrix are the total costs of the product departments under the allocation scheme used herein. These costs will be used in the next chapter as a basis to determine the cost of the products of the schools and col— leges of USC which produce accredited college-level educa- tion. CHAPTER IV COSTS OF DEGREES Purpose The purpose of this chapter is two—fold: first, the individual input costs of the several degrees awarded by USC will be determined; second, the average costs of the bacca— laureate and master's level degrees will be calculated. The first purpose supports the second purpose; each purpose requires several preparatory steps. There are five steps necessary to accomplish the first purpose. First, the outputs of the USC Product Depart— ments will be determined in terms of course hours. Second, these outputs will be weighted to permit expression of them as equivalent units. Third, the cost per equivalent unit will be calculated and converted into a cost per hour. The fourth step will be to determine the number of credit hours by type that comprise the various degrees. Fifth, the costs of degrees will be calculated, and the reliability of the results will be determined. The first five steps will determine the average cost of a bachelor's degree and the average cost of a graduate degree for each of the schools and colleges at USC. The 44 1.4 45 specific costs of the individual degrees which generate these average degree costs will enter into the computation of the costs of an average or composite undergraduate and an average graduate degree on a university-wide basis. To calculate the costs of the university—wide com- posite or average degrees for both the graduate and under— graduate level requires the use of random stratified sam- pling techniques for two reasons. First, there is wide variability in the number of degrees awarded at the same level by different colleges and schools. Second, there is wide variability in the costs for degrees at the same level from the different colleges and schools. Use of a simple average (mean) cost of the Specific degrees fails to take into account that there are more degrees awarded at one cost than another: use of a simple random sample fails to take into account that a disproportionate number of degrees may be drawn from the Specific degree types. There are three steps required to accomplish the second purpose because the statistical data developed in accomplishing the first purpose must be rearranged to pro— vide the basis for application of the stratified sampling technique. First, the stratified sampling technique will be introduced. Second, this technique will be applied against the unit cost data developed by the first five steps. Third, the results will be stated. 46 Product Departments' Outputs Background The outputs of colleges and schools constitute a product line of almost infinite variety. Research, publica— tions, public services of an administrative and advisory nature, plus student education are examples of this variety. This dissertation, however, concentrates on but one specific product—-accredited semester hours of successful work--to the exclusion of all other outputs. This concentration 1 requires a nonempirically based judgment of the percentage. of each product department's effort that is devoted to teach— ing. This is a nonempirically based judgment because almost no objective detailed data are available relative to the use of professorial time.1 The need for such objective data is well known and detailed in Doctor Jones' 1964 study for U.S. Office of Education.2 Until research on this subject is com- pleted, this and other studies must rely on the best non~ empirical opinions available relative to the disposition of an educational institution's output. lGardner M. Jones, "A Procedural & Cost Analysis Study of Media in Instructional Systems Development," Part B. Office of Education, Department of Health, Education and Welfare, 1964, p. 103. 21bid. 47 Product Input Value Colleges and schools produce such a wide variety of services that some procedure is required to assess the value of any one service. The procedure used in this work is to determine the input value of teaching to the exclusion of the other products. The input values of the total products of the colleges and schools of USC have been determined in the T Matrix of Table 2 and will be used in Table 5 below. However, this T Matrix does not include capital charges such as land rental, building depreciation and the like.7 These capital costs are subject more to local conditions than to general economic circumstances whereas other university costs are more subject to general economic conditions. This is because the Specific location in an urban center as compared to the suburbs greatly affects the value of land and build— ings both in exchange and as a tax basis. Other university costs are incurred to acquire educational and professional equipment, the services of educational and administrative staffs, etc. Such costs items are taken from regional and national markets and therefore will not vary as much from one school to another as will capital costs. As a result, the elimination of capital costs makes this work more gen- erally applicable. As previously stated the costs of each product department's output is contained in the T Matrix. The value of the teaching output is that percentage of a college's or 48 a school's output devoted to teaching. To determine this percentage, the deans, acting deans, assistant deans and other senior and knowledgeable people in each college and school were interviewed. In these interviews two things were sought: first, the percentage of effort going into teaching; second, the relative weights of the various courses. This second interview objective will be treated later. The results of the interviews relative to the per- centage of effort devoted to teaching are given in Table 3. Table 3. Percentage of total effort devoted to teaching College or School Percentage Arts and Science 90 Business 80 Education 90 Engineering 65 Journalism 75 Law 80 Nursing 85 Pharmacy 90 These data will be incorporated in calculations later in this chapter and are less objective than if there were empirical data available relative to the disposition of professorial time. 49 Equivalent Units of Output Weighting of the various courses offered by the Product Departments permits all courses to be eXpressed in terms of an equivalent unit. To accomplish this, courses were divided into graduate and undergraduate levels and further subdivided into laboratory and nonlaboratory ele- ments. This four-way weighting is relevant to the situation at USC although it is clearly recognized that other weight— ing systems may obtain for situations at other universities. For example, some universities may be such that courses should be divided into upper, lower, master and doctoral levels; then further subdivided into laboratory, nonlabora— tory, large lecture, television, etc., elements. The results of the subdivision system employed in this work are given in Table 4. Table 4. Course weights* Course Type Undergraduate Graduate College or School Lab. Non-Lab. Lab. Non-Lab. Arts and Science Business Education 1 Engineering Journalism Law Nursing Pharmacy H+4c3m14c>o+a Hrac>m+ac>wra orac>o<5\ioc3 OF4F4©t0\Jk)N *The value 0 indicates no such course available. 50 Table 4 eXpresseS the weight ratio between courses of the same college but not among courses of the various colleges. For example, in the case of the School of Jour- nalism the ratios of 6, 8 and 9 express the opinion of the Dean of the School of Journalism that graduate work in Journalism requires 150 per cent of the effort required for undergraduate work if the course does not involve a labora- tory session. If the course involves a laboratory session the effort required is increased by one—third. These in— j sights were then converted to ratios containing only the whole numbers 6, 8 and 9, to expedite calculations of unit cost of output. The numbers are relevant only to Journalism. Cost per Equivalent Unit Output of Product Departments The semester hours of successfully completed col- lege-accredited work at USC are recorded in the central data bank of USC. These data are so arranged that it is possible to identify semester hours by the college or school of their origin, their department and their Specific class section but not by teacher. A computer program was written to retrieve this information for the two summer sessions of 1966, the fall semester of 1966, and the Spring semester of 51 1967.3 This time period coincides almost exactly with the fiscal period for which costs are relevant. In actuality the teaching calendar begins approximately two weeks before the fiscal calendar. Both the chief accountant and business manager of USC state that this negligible mismatch has no measurable or meaningful effect. The computer program for retrieving these data generates the number of hours of work attempted, failed, and completed. Only successfully completed hours are treated as product. This is consistent with the normal practice of not considering rejected items as output. Unsuccessfully attempted semester hours are considered to be in the same category as unacceptable industrial product, that is, they are part of the cost of producing the useful products. Calculation of Cost_per Equivalent Unit The successfully completed semester hours of each product division of USC are multiplied by the weighted values in Table 4. This multiplication converts the successfully completed hours of each course into equivalent units of out— put. These units of output are then totaled for each school or college to accumulate the total number of equivalent 3This computer program is relative only to the storage routines used by the USC data bank or to identical systems. Because this severely limits its general applica— bility this program is not reproduced in this work. 52 units delivered. The results are arranged in Column c of Table 5. The Column (d) of Table 5118 determined in a straight- forward manner. For example, 90 per cent of the $7,483,885 input value of the College of Arts and Sciences is devoted to producing the 169,783 equivalent units of output. This yields a cost per equivalent unit of: $7,483,885 169,783 x 0.9 = $39.67 (4-1.0) Table 5. Equivalent units Column (a) (b) (c) (d) _ . Table of Cost per College or T Matrix Table 3 Equivalent Equivalent School Value Value Units Unit Arts & Science $7,483,885 0.9 169,783 $39.67 Business 1,582,601 0.8 41,358 30.62 Education 1,001,605 0.9 170,288 4.705 Engineering 800,977 0 65 7,922 65.72 Journalism 174,747 0.75 14,952 8.765 Law 361,019 0.80 6,775 42.63 Nursing 311,018 0.85 3,819 69.22 Pharmacy 221,718 0.90 3,857 51.74 Unit Cost Matrix Table 5 lists the equivalent unit cost for each col— lege or school and Table 4 lists the course weights. Multi- plication of the equivalent unit cost by its course weight yields the input cost per semester hour of delivered 53 instruction. The results of this multiplication are given in Table 6. In this table only one listing is made for each nonzero course weight of each school. This table is designed to identify the 14 x 1 matrix which will be used later. The unit cost of output having been determined and arranged into the matrix format desired, the next step is to determine the number and types of semester hours which com— prise the delivered education upon which awards of various degrees are based. Table 6. Semester hour cost matrix 14 x 1 Matrix College Equivalent Equivalent of Semester or School Course Typea Unit Weight Unit Cost Hours Costs Arts and UGNL & L 1 $39.67 $39.67 Science GNL 2 39.67 79.34 Business UGNL l 30.62 30.62 GNL 2 30.62 61.24 Education UGL & NL 10 4.705 47.05 GL & NL 17 4.705 79.99 Engineer- UGL & NL 1 65.72 65.72 ing GNL 2 65.72 131.44 Journalism UGNL 6 8.765 52.59 UGL 8 8.765 70.12 GNL 9 8.765 78.88 Law GNL l 42.63 42.63 Nursing ALL 1 69.22 69.22 Pharmacy UGL & NL 1 51.74 51.74 aCode: UG = Undergraduate; G = Graduate; NL 2 Non— laboratory; L = Laboratory. 54 Determination of Degree Composition Background Degrees are generally awarded for completing a specified number of courses and for attaining a grade point ratio above some minimum point. During any period a univer- Sity delivers many courses with various numbers of hours of credit granted for these courses. In terms of degrees, a university's output could be measured in equivalent degree units. For example, if a school produced 120 semester hours , of successful teaching during a period, and the degree requirements of that university were 120 semester hours, the equivalent unit of output would be one degree. Thus a por- tion of the product of a university can be measured in terms of degrees. Within a university it is common for the require— ments for different degrees to vary. For example, some degrees at USC require 120 semester hours whereas other degrees require 128 hours and still other require a differ- ent number of credit hours. This variable requirement of semester hour totals is not the only variable. Degree requirements also vary in the type and mix of courses required. AS a result, one degree may differ very much from another in both the number of hours and in the type of hours required. This variability in the number and type of required courses necessitates that statistical sampling 55 techniques be used to identify the ingredients and therefore to calculate the costs of representative degrees. Statistical Population The population upon which the statistical measure- ments of this dissertation are based is taken from the USC roster of degrees awarded for the period under study. The number of these various degrees awarded is given in Table 7. These degrees, which were awarded during the fiscal year 1967, are to a great extent based upon teaching deliv— ered in prior years. This raises the question: Should not the costs of degrees be based upon the costs that pertained at the delivery date of each course comprising the basis for awarding the degree? The answer posed in this work is: The current going rate of cost is sought, not the historic cost. The purpose of the next section explains the statistical sampling technique used to determine degree costs. Sampling Technique The statistical portion of this dissertation relies upon the work of W. G. Cochran4 and Taro Yamane.5 The popu— lation from which the samples were drawn is taken from the 4W. G. Cochran, Sampling Techniques (2d ed.; New York: John Wiley & Sons, Inc., 1963), pp. 93-118. 5Taro Yamane, Elementary Sampling Theo£y_(Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1967), pp. 64—158. 56 Table 7. USC degrees awarded, fiscal year 1967 Degree Awarded Number of Degrees Sample i1ze (Nh) (nh) Bachelor's Level: Arts & Sciences 390 39 Business 352 36 Education 226 23 Engineering 79 8 Journalism 33 5 Nursing 40 5 Pharmacy 49 5 Totals (N) 1,169 121 Master's Level: Arts 74 9 Business 58 6 Education 31 5 Engineering 16 5 Sciences 59 6 Totals (N) 238 31 Graduate Level: Law 96 10 *For convenience, Nh, the population strata of Popu— lation N, and nh, the sample size used later in this chapter are identified in Table 7. rosters of the three USC graduations which took place during fiscal year 1967. The data in Table 7 were compiled from these rosters. The three rosters were reassembled in such a way that rosters of the recipients of Similar degrees, say a Bachelor of Arts, were butted against each other to make one composite roster. Therefore, each such composite roster contained three alphabetized listings sequentially arranged. 57 These rosters were then subdivided into units containing 10 names each to ensure a minimum 10 per cent sample. A table of random numbers was then entered at a random point, and the list of random digits that followed was used to deter— mine which name in each group of 10 names Should be included in the sample. The first random digit would indicate which of the first ten names to use, the second random digit the name in the second decade, etc. The transcripts of the selected graduates were then drawn from the record file of USC and the semester hours of work accomplished by those students were systematically tallied. Hours were categorized in a system compatible with the tabulation Shown in Table 6. .As a result, each tran— script generated a l x 14 matrix element in which the row value was the number of hours completed in each of the 14 "course types" of Table 6. This arrangement generates an n x 14 matrix, where n is the sample Size. This arrangement also facilitates additional computations using the 14 x 1 matrix of Table 6. These calculations determine the costs of degrees. Degree Costs and Reliability Degree Costs The cost associated with the delivery of the instruc— tion necessary for awarding a degree may be obtained by add— ing the products of multiplying the amount of each type of 58 instruction by its appropriate cost. The amount of instruc— tion has been tallied from n transcripts for each of the 13 types of degrees shown in Table 7 above. This results in a series of 13 matrices that are n x 14 in dimension where n, it will be recalled, is the sample size. ,For example, 39 of the 390 transcripts of students who were awarded bachelor degrees in Arts and Science have been randomly selected. The first transcript selected generates the first row of the 39 x 14 matrix of bachelors' degrees in.Arts and_Science. Each of the 14 columns in this row indicates the number of credit hours taken in the matching "Course Type" column of Table 6 in the precise order shown in Table 6. For example, the first column of the 39 x 14 matrix is Arts and Science course type UGNL & L; the second column in Arts and Science Course type GNL; etc. Continuing this process for each randomly selected transcript from each type of degree of the 12 remaining results in a series of 13 matrices, each of which is n x 14 in dimension, where n is the sample Size shown in Table 7. Each of these matrices was then multiplied by the 14 x 1 matrix of Semester Hour Costs of Table 6. Because the sum of the n'S was 162 and this results in a very burden- some series of 162 X 14 calculations, this operation and the statistical computations were accomplished on a computer. The results are given below in Table 8. 59 «HN.H mmmm oma Ham mmmm 3mg u H ®>®1H THMSCMHU mmm m oeem mom owe seem mmosmaom meHHH.omes oae Hmsa owes mcaummcamcm 44m H.mmom mom mas mmom acanmosom omm.H some 4mg oom coma mmmaamsm 084 + moem Nam emm moem muse "He>eq m.HeummE eve m emme «me mom emee Suwaneem mm H.4mme om 6mm 4mme maumusz wee.H omem mmm eem omem amaamcusoe mo¢.H.emoe mam ems emee acaummcamam mma H.eamm me omm eamm nonsmosem mmH.H moms man awe moms mmmaamsm mm + Amman «m w Hamw Hmmsw moamaom a mpu< NSmY N "HON/mg m. HOH mfiomm Hm>mq mosmpflmsoo HOHHM seebme>mn umoo mesmem petumBfi meummm bceo Hem om pumpcmum Unmecmum mmmue>< mo smmz mmmummp we umoo .m magma 60 The computer program which performs the calculations which result in Table 8 is given in Appendix III. This pro— gram besides performing calculations which indicate the reliability of these results also calculates some data to be used in the stratified sampling calculations of a later portion of this chapter. Reliability of the Results The reliability of the statistical sampling is shown in Table 8. A 90 per cent confidence interval is given for each of the types and levels of degrees awarded. This was obtained by multiplying the Standard Error of the sample mean by the value of the standard Normal Deviate correSpond- ing to 90 per cent of the area under the normal curve. The intervals for various degrees of confidence can readily be obtained by extracting the appropriate "Normal Deviate" given in any "Areas under the Normal Curve” table for the confidence desired. The 90 per cent confidence interval in Table 8 may be taken to mean the following. If another sample of n Size, say 39, were randomly drawn from the population, say the 390 graduates who were awarded a bachelor's degree in Arts and Science, there is a 90 per cent probability that the average cost of delivering the instruction necessary for that degree would be between $4825 and $4937. There is a 10 per cent probability that the average cost would be less than $4825 or greater than $4937. 61 The Stratified Costs of Degrees The average cost of an undergraduate degree in each of the colleges—-Arts and Science, Business, etc.--has been determined, as has been the average cost of each of the various graduate degrees. However, the costs of the average undergraduate degree at USC as a whole and of the average graduate degree have not been calculated. Determination of these costs requires further analysis. The graduate degree in Law needs to be eliminated from further consideration for two reasons. First, this degree is currently (at the time of this study) being changed from a Bachelor of Laws to a Doctor of Laws degree. All other graduate degrees in this study are scheduled to continue at the master's degree level. Second, the educa— tion delivered to qualify students for a law degree are peculiar to the Law School and requires two years rather than one year of study. After elimination of Law from further consideration one additional calculation is worthwhile. Thus far we have the mean cost and a confidence interval for each of the various degrees awarded by USC. We have no mean cost nor confidence interval for the average USC degree awarded at either the bachelor's or master's level. Further, the divi- sion into specific degree types of the populations of both master's and bachelor's level degrees has resulted in some very small subdivisions or strata. This division has 62 resulted in some very high and some very low costs for the specific degree types. A Simple random sample of either population could lead to a less efficient estimate than other sampling routines because of this variation within each population. Stratified sampling mitigates these cir- cumstances. The Stratified Sampling Technique Stratified sampling is a method which provides a more efficient estimate of the real mean of a population (than the simple random sampling) when the population can be stratified into homogeneous sub-populations. In essence the stratified sampling technique takes a sample from each of the various strata which composes the overall population.6 The design of this work is to take from each stratum a sample whose size is proportionate to the size of its stratum. The overall population which encompasses the various strata is the roster of degrees awarded by USC during 1967. The USC graduates are divided into one population of those receiving undergraduate degrees and another popula- tion of those receiving graduate degrees exclusive of Law. These two populations are next separately arranged into one stratum for each type of degree awarded. By previous design 6Stratified sampling is a well-defined technique covered in many standard texts. .As previously stated, this dissertation relies upon the works of Cochran and Yamane cited above. 63 the sample Size selected and used in the first sampling pro- gram was the greater of 10 per cent of each stratum (degree type) or 5 in order to avoid the chance that a smaller sam— ple might not be as representative of the population. The computer program in Appendix III contains instructions for the appropriate data to be calculated. The statistical terms used in defining these values and the values given in Tables 7 and 8 are listed in Table 9. Table 9. Table of statistical terms N = Number of degrees in a population Nh = Number of degrees in stratum h nh = Sample Size for stratum h §'= Estimate of the mean of the population Xh = Estimate of the mean of stratum h sh — Estimate of the variance of stratum h <>==Estimate of the variance of x S ='4 A = Estimate of the standard error. Application of the Statistical Technique Computation of the estimated average cost of an undergraduate and of a graduate degree is done by:7 7Cochran, op. cit., p. 89. The symbols in Cochran's equation (5.1) are modified to make it consistent with terminology used herein. 64 (4—2.0) Computation of the estimate of the variance of i’is obtained by:8 2 h h [D ___L_ _ V(x) ‘ N2 2 Nh(Nh nh) n (4—3.0) The values for Nh and nh are given in Table 7; the remaining values are given in Table 10. Solution of equations 4—2.0 and 4-3.0 yields: §'= $5290 for undergraduate degrees s": $2925 for graduate degrees ‘0’: $1869 for undergraduate degrees \O/= $10,954 for graduate degrees 8 = $43 for undergraduate degrees S = $105 for graduate degrees. Statement of Results The results of the Stratified Sampling Technique as applied against the statistical data of this dissertation permit the following statement. The average input cost of the education required for the awarding of a bachelor's degree is, with 90 per cent 8Cochran, op. cit., p. 93. Cochran's equation (5.11) modified for consistency with present terminology. 65 0:5 cans pcmummmae mauemaam mum .HMHHOU pneumes ebb 0b pepsson seen e>m£ hes» m mHQmB SH emsmomfl m eabme SH peumea mmSHm> am we mesam> exp we been eumsqm wees N mseomeomo emommmoaa emam memmm Haeaam mm mmocmaom mmmmeoae men mommos maomaom Ha meaummcamam mmmmemmm mom meomm memmea em coapmosem Geomemme eaom aooma mooom mm mmeamsm ommmmemem cams maoee mmavme me mesa "Hm>eq m.bmummz Hammsmsmmm mowowemmm emam moomea oaommm as somsumem ooemmv oosa sum meme mm maumusz ememesae vmm ammoe msemmm mm amaamaH50b Homemmmom moem mmmmm mamame He maaummghmcm seeommssN memmw mmmm mmemma mom nonsmosem mmmmmmmmaa mmmaaa eemma Hmmmow 64m mmmcamsm omommeema ommoma mean amass Hmm mogmaom a mama nam>eq m.HOHm£Umm as..: a as..n : :c\ gm em as..n nc\memx zv z A zv z m 4m 2 memumev mo mumoo peHMHumHum .OH mHQmB 66 confidence, between $5,219 and $5,361. The average input cost for the education required for the awarding of a master's degree is, with 90 per cent confidence, between $2,752 and $3,098. CHAPTER V SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Purpose The purpose of this chapter is three-fold. First, the research methodology and results will be summarized. Second, conclusions will be drawn relative to other research methodology and results. Third, recommendations will be made with respect to possible use of the research results and future research of a Similar or related nature. Summary This research divided the on—campus organizations of USC into product or service departments. The services ren— dered by the service departments, as measured by their input costs to USC, were allocated to the product departments. These allocations were based on the amount of space occupied, actual expenditures, student load and the opinions of the most knowledgeable personnel available when more empirically oriented grounds for allocation did not exist. These alloca— tions were made by using the Gauss-Jordan matrix inversion technique as adapted by Williams and Griffin,1 for cost 1T. H. Williams and C. H. Griffin, op. cit. 67 68 allocations. The computer program for this matrix inversion routine is Shown in Appendix II—l. After the allocations of the costs of the service departments to the product departments were determined, only those product departments which provided instruction for col- lege accredited courses were given further consideration. These product departments--eight on-campus schools and col- leges—-are producers of research, publications, services and education; of these products only education was dealt with further. This was accomplished by dropping that portion of ,i each product department's total cost not ascribable to educa- I tion as determined by the judgments of the most knowledgeable personnel available. These steps produced a figure identi- fied as the total cost, net of capital charges, of the educa- tion produced by the eight schools and colleges. It did not, however, measure, what education per se had been produced. Education output was measured, initially, in terms of successfully completed semester hours of instruction. This was accomplished by extracting from USC'S data bank the total number of semester hours by course and section. Each course was then assigned a weight proportionate to the effort required to deliver that course. In making such weight assignments it was again necessary to rely on sub— jective evaluation made by the most knowledgeable persons in that area. Multiplying the weight so derived by the number of relative semester hours permitted eXpression of the 69 entire output of each school and college in terms of equiva— lent units. Division of total costs by total output yielded the cost per equivalent unit of instruction: multiplication of the cost per equivalent unit by its relative course weight yielded the input cost per semester hour. Multiplica— tion of the semester hours included on the transcript of a degree awardee, by the appropriate cost per semester hour yielded the input cost at USC for the education upon which award of degree was based. Sampling techniques were selected to determine the , cost of representative degrees by type and level. The sta— tistical technique employed entailed a randomly selected sample, of either five cases from the population or 10 per cent of the population, whichever was larger, upon which to calculate the estimated mean and confidence intervals for each degree type and for each degree level. These calcula- tions were accomplished by computer, the computer program for which is given in Appendix IV. The estimated costs of degrees by type attained by use of these statistical techniques are Shown in Table 10 and rearranged in a descending cost order by degree type in Table 11. The estimated average cost of degrees by level was found to be, with 90 per cent confidence, between $5,219 and $5,361 for undergraduate degrees and between $2,752 and $3,095 for master's level degrees. 70 Table 11. Cost of degrees CoSt Rank 90 Per Cent Confidence Level Bachelor's Level: Pharmacy 1 $7687 i 747 Engineering 1 7687 i_ 408 Nursing 2 7254 i_ 33 Journalism 3 5780 i 474 Education 4 5317 j_ 123 Arts & Science 5 4881 i. .56 Business 6 4508 i. 189 Master's Level: Engineering 1 $4480 : 1168 . Education 2 3098 i_ 344 Sciences 3 2776 i 339 Arts 4 2708 i 480 Business 5 1960 :_ 220 Conclusions Background 1 Conclusions based on the work accomplished in the dissertation need, at the outset, to be circumscribed by limitations imposed on the research. Afterward, tentative conclusions can be drawn from the research. Research Limitations The research in this dissertation has one basic limitation imposed upon it which is worthy of elaboration. This limitation is the lack of empirically based data upon which to make allocations. 71 Within this general limitation there are two spe- cific areas which may be seriously affected by the lack of empirical data. In the first instance, some allocations of service departments' outputs had to be based upon the con- sidered opinions of the most knowledgeable personnel avail- able. In the second instance, the allocations of the pro- ducing departments were similarly based upon opinion. In the later case, the proportion of professorial time devoted to teaching, as compared to time used in re- search, writing and providing other professional services, , I could not be precisely allocated. Allocations by Nonempirical Judgments Various allocations were made throughout this disser- tation. In some instances allocations had to be based upon the judgments of the most knowledgeable personnel available and as such are the best available data. As previously noted, such allocations may be affected, among other things, more by recent experience and circumstances surrounding the individual person making the judgment than by the average or normal experiences and circumstances. Therefore, such judg— mental allocations may be less objective than allocations based upon empirical data. Professorial Time The total teaching costs of the product departments of USC were spread over the total number of successful 72 semester hours of delivered instruction by means of the equivalent unit. These hours of instruction were summed from the total number of hours delivered by each section of every course. Among these sections there existed variations in the number of students and the salaries of the teachers. Because no existing records were detailed enough for more penetrating analysis, this dissertation could not consider the cost differences that exist, from section to section and from course to course as section size and professorial sal- aries vary. For example, in this work, the hours of educa- tion produced by a small section taught by a highly paid faculty member would be costed at the same hourly rate as a large section taught by a low-paid graduate student, and vice-versa. Although there will be some averaging out of semester hour costs because so many courses, sections and faculty members are involved, clear probability exists that highly paid senior faculty members will tend to teach the most advanced courses and lower paid junior faculty members or graduate assistants will tend to teach the introductory courses. The sections of the more advanced courses tend to be smaller than those of the introductory courses. As a result, a larger proportion of the costs of professorial salaries should be allocated to some sections and courses than to others. In this dissertation, these salary costs have been allocated as though all sections were homogeneously 73 constituted and taught. AS a result, the costs of lower level courses may be overstated and those of higher level courses may be understated. This tendency will be offset in part by the heavier weight given to graduate courses as reflected in Table 4. It will be offset further because each degree requires a mixture of courses from various levels. Thus, the understated costs of the higher level courses will be offset by the overstated costs of the lower level courses. The computed degree costs should, therefore, serve as reasonable estimates of the real input costs of degrees. Matrix Algebra Techniques The matrix inversion technique and the matrix multi- plications used in this dissertation were adaptable to com- puter usage. The inversion of the A Matrix——the cross— allocations of the service departments' outputs among ser— vice departments——when done by the computer required slightly over two minutes. This inversion required solution of 40 Simultaneous equations in 40 unknowns. The approximate num- ber of mathematical operations involved in this procedure on a matrix of this size can be estimated by taking into consid- eration the machine time and the machine speed. The computer performs an operation in 8 microseconds: it performs 120,000 operations per second. Use of machine time rounded to 120 seconds, multiplied by the rate of 125,000 operations per second yields 14,000,000 operations performed. If a manual lllllllllllll 74 operator could perform one operation per second, without human error, this calculation would require approximate 2.1 man years of 2,000 hours each. Thus matrix algebra tech— niques when performed on a computer and used to allocate overhead free enormous amounts of accounting time which can be devoted to refining the basis of the cost allocations or for other purposes. Recommendations The recommendations made in this work fall into three categories: first, a group of recommendations dealing with the pricing of education; second, a group of recommenda— tions pertaining to the use of the techniques employed in this work; and third, a group of recommendations treating with the need for further research related to this work. Pricing of Degrees At USC and many other educational institutions the price of degrees—-tuition——is invariant with respect to the type of degree sought by the student. This work demonstrates that the cost of producing all USC degrees is higher than the tuition charges of $500 per year for in—state and.$1,000 per , year for out-of—state students. For example, the bachelor's degree with the lowest input cost is in business for a total of $4,508, net of capital costs. The tuition charged for four years is $2,000 in-state and $4,000 out-of—state. The master's degree in business has the lowest input cost, 75 $1,960, net of capital costs, and the tuition charge for it is $750 in-state and $1,500 out-of—state. The market value of degrees varies dependent upon earning opportunities and prestige associated with the fields of endeavor for which given degrees qualify their recipients. The value of degrees to their holders Should be researched. .Where the market value of a degree exceeds tuition consideration should be given to increasing tuition by the amount by which the input costs of the degree exceeds the tuition presently charged. This work does not recommend, I per se, that there should be a different tuition rate for each of the degrees available. It recommends that Since a cost difference exists, the value-to-holder difference may be a justifiable basis upon which to introduce a price dif- ference to meet the need for additional funds or to estab- lish some parity between value given and received. Social Welfare Based Pricing There are some degrees whose input costs may exceed their market value as well as the tuition charged for them. Some of these degrees may have a high social welfare value; for example, nurses provide a service that is essential to society in general. It may well be that irrespective of the market value of a nursing degree it is in the best interest of society to subsidize such training. 76 This work recommends that, where possible, some assessment be made of the social welfare value of the impor- tant services that degree holders provide to the general community. Administrators Should, therefore, take into account the input cost, tuition price and value of degrees-- both to the community and the degree holder and consider adjusting prices-—tuition rates--to encourage students to undertake studies in areas which provide needed services to the larger community. Out-of—State Tuition Rates The average cost of either an undergraduate or a graduate USC degree exceeds the price paid for the education upon which it is based. For example, the out—of—state tui— tion at USC is $1,000 per two—semester academic year. The average undergraduate degree has an input cost of between $5,219 and $5,361, with 90 per cent confidence. An out—of- state student, therefore, receives a subsidy to the extent that the input cost of his degree exceeds the tuition charged him for his education. Administrators shouhfl consider whether out—of—state tuition rates should cover at least the input costs of delivering the education to the recipient. In particular, institutions like USC which are publicly sup- ported, Should take into consideration the number of out—of— State students it and other state institutions educate from each state and the number of students from its state that are being educated in each of those states. The relative 77 differences between costs and prices of each such institu- tion Should be compared to USC'S cost-price differential to evaluate whether quotas should be imposed against states who seem not to support "fairly” public education. To do other- wise could be interpreted as a subsidization of other states by USC or vice-versa. Research Techniques There are sizeable amounts of accounting and man- agerial time made available by the matrix inversion and matrix multiplication techniques used in-—but not originated by--this dissertation. Further, these techniques are read— ily adaptable to other conglomerate type organizations where allocations of overheads to ultimate products is abstruse. This technique should be adopted in such cases. Part of the time saved could be Spent in researching for better input data upon which to build a set of empirically based alloca— tion decision rules. Further Research Cost allocation is usually based upon direct labor. In an educational institution professorial time is the direct labor element. Yet little if any empirical evidence exists relative to the way this time is devoted. Profes— sorial time goes into research, administration, writing, teaching, and other related activities. What is needed is a definitive examination of professorial time utilization. 78 The work in the dissertation leads to the clear conclusion that detailed research Should be done on statistically significant samples of professors at a large number of schools. 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