## Abstract

We describe a performance budget planning model developed for a research university, comprised of a set of 88 key variables and 38 non-linear structural equations that describe interactions among them. These equations, based on the knowledge of research university’s financial working and theoretical considerations, relate expenditures and revenues to teaching and research operations. We demonstrate the value of this model for developing insight into the financial structure of the university. In particular, we show how the model aids in (1) comparing the effect of various policy alternatives on the performance of the university, (2) performing comparative statics analysis of any subset of variables of interest, (3) choice of policy variables and policy alternatives, and (4) gaining insight into the structure of the interactions for a given policy alternative in terms of the causal chain between policy variables and outcome variables. We also describe a computer implementation of the model and discuss a class of mathematical tools for policy planning analysis that facilitate the use and manipulation of models based on sets of nonlinear constraints.

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## References

Hood, W. C., & Koopmans, T. C. (eds.) (1953).

*Studies in econometric method. Cowles commission for research in economics*. Monograph No. 14. New York, NY: Wiley.Koopmans, T. C. (Ed.) (1950).

*Statistical inference in dynamic economic models*. Cowles commission for research in economics. Monograph No. 10. New York, NY: Wiley.Simon, H. A. (1953). Causal ordering and identifiability. In:

*Hood and Koopmans*[1], chap. III, pp. 49–74.Simon, H. A. (1967). The job of a college president. In:

*The educational record*(pp. 68–78). Washington, D.C: American Council on Education.

## Acknowledgements

The foundations of this paper were laid by Herb Simon. His are all the equations in the model as well as the idea of building strategic planning systems based on structural equations and capable of making explicit the causal ordering that they imply. Herb’s untimely death in 2001 has prevented us from publishing this paper jointly. We would like to thank Dr. Velupillai Kumaraswamy and an anonymous reviewer for suggestions that led to revising of the paper. Dave Zubrow, Jeff Bolton, Kevin Lamb, Felicia Ferko and Igor Reshetar from Carnegie Mellon University’s Office for Planning and Budget helped us with the development of PBM. Hong Shi, a graduate student in Information Science, University of Pittsburgh, helped with the programming end of an initial version of the system. The graphical user interface to PBM has been implemented in GeNIe, a graphical modeling environment originally developed at the Decision Systems Laboratory, University of Pittsburgh, available at http://www.bayesfusion.com/. We were supported in this work by a special Grant from the University Administration at Carnegie Mellon University. The first author was additionally supported by the National Science Foundation under Faculty Early Career Development (CAREER) Program, Grant IRI–9624629 and by University of Pittsburgh Central Research Development Fund and, more recently, by the National Institute of Health under Grants U01HL101066-01 and 1R01HL134673-01 and Department of Defence under Grant Number W81XWH-17-1-0556.

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## Appendix: PBM Variables and Equations

### Appendix: PBM Variables and Equations

The 88 model variables are divided into seven submodels: *Teaching Operations*, *Teaching Expenditures*, *Research Expenditures*, *Income*, *Space Cost*, *Total Expense*, and *Surplus*, each described in detail in the following sections. We would like to point out that some equations describe interactions among variables within a submodel. Others, involve variables from two or more submodels. Where an equation of the latter type will be represented in a causal graph depends on the causal ordering of the variables and this, in turn, which 50 of the 88 model variables are exogenous. Faced with the task of structuring our presentation so as to maximize clarity, we decided to discuss each of the equations within a submodel and chose the most typical submodel that it relates to. To help the reader navigate through the model, we report for each model variable the equation number where it appears.

### 1.1 Teaching Operations Submodel

The *Teaching Operations Submodel* relates student and faculty numbers to class size, teaching loads, and student–faculty ratio. The variables included in this model are: *studu* (1, 6, 22), *studg* (1, 2, 5, 6, 21, 22), *stud* (1, 4, 6), *tasst* (2, 4, 12, 21), *rasst* (2), *facten* (4, 5, 10, 11, 18, 19, 20, 28, 29), *facspec* (4, 10, 11), *tloadsp* (4), *tloadg* (3, 5), *tloadu* (3), *tloadasst* (4), *tloadten* (3, 4, 7, 8), *credhrs* (4, 5, 7), *flload* (8), *tchfrac* (8, 10, 11, 18, 19), *clsze* (4, 6, 7), *clszeg* (5, 6), *clszeu* (6), *ratio* (7).

The total number of students (*stud*) is the sum of the total number of undergraduate students (*studu*) and the total number of graduate (doctoral) students (*studg*).

The following equation ties the number of teaching assistantships (*tasst*) and the number of research assistantships (*rasst*) to the total number of graduate students (*studg*). It expresses the assumption that all doctoral students hold either a teaching assistantship or a research assistantship. (The variable *rasst* is not used in any other equation. However, it is used indirectly in Eq. 21 through the term (\({ studg} - { tasst}\)).

The average teaching load for tenurable faculty (in terms of semester hours per week) (*tloadten*) is the sum of the load of undergraduate (*tloadu*) and graduates (*tloadg*) courses,

Class size (*clsze*) is expressed by the number of students (*stud*, this is multiplied by the average credit hours taken, *credhrs*) per semester credit hour taught (this is expressed in terms of the numbers of tenurable faculty, TAs, and special faculty times their respective teaching loads),

Average graduate class size (*clszeg*) is expressed as the total credit hours taken by graduate students (\({ studg} \times { credhrs}\)) divided by total semester hours taught by regular faculty (\({ facten} \times { tloadg}\)). (Note: This implies that only regular faculty teach graduate students.)

The student to faculty ratio (*ratio*) is the ratio of total credit hours taught by tenurable faculty (their teaching load, *tloadten*, times the average class size, *clsze*) and the total credit hours taught (by all faculty and TAs),

The average fraction of tenurable faculty time allocated to teaching (*tchfrac*) is simply expressed as the ratio of the actual teaching load of tenurable faculty to a hypothetical full-time teaching load (*flload*) that assumes only modest research activity,

### 1.2 Teaching Expenditures Submodel

The *Teaching Expenditures Submodel* describes expenditures for instruction. The variables included in this model are: *frng* (9, 12, 20), *tuitg* (12, 21, 22), *salten* (10, 18), *salspec* (10), *ovrhdt* (9, 11, 12), *ot* (11), *exptch* (12, 37), *exptdir* (9, 10, 12), *ovrhdtamt* (9).

Only Eq. 9, computing the dollar amount for overhead on instruction (*ovrhdtamt*) from the total direct teaching expenditures (*exptdir*), describes internal interactions within the *Teaching Expenditures* submodel

Equations 10, 11, and 12 describe interactions between the *Teaching Operations* and *Teaching Expenditures* submodels.

Direct teaching expense (*exptdir*) is calculated as the sum of the salaries for special teaching faculty (\({ facspec} \times { salspec}\)) and the portion of regular faculty assigned to teaching by the teaching fraction term (*tchfrac*). (Note: The salaries for teaching assistants are not included as a direct expense, so they are automatically excluded from the base upon which overhead is computed.)

The following equation calculates an overhead rate for teaching (*ovrhdt*) which is based on the fraction of tenurable faculty time allocated to teaching plus special faculty (full time). (Note: TAs are excluded from this calculation.)

The following equation computes total teaching expenses (*exptch*). Overhead is applied to total faculty compensation (salaries plus fringe). TA salaries ($12,000 per annum) are added to direct and indirect expenses for regular and special faculty. This assumes the same fringe rate for regular and special faculty.

### 1.3 Research Expenditures Submodel

The *Research Expenditures Submodel* contains equations that determine expenditures for research. The variables included in this model are: *exprdirru* (13, 15, 17, 20), *ovrhdrru* (15, 17, 20), *exprdir* (13, 16), *ovrhdramt* (16, 17), *ovrhdramtac* (17), *expresru* (14, 15), *salgradpc* (21), *mo* (18), *facres* (18, 19, 28, 29), *salres* (18), *facpcsub* (18), *expdoc* (21, 37), *exprdirac* (13, 18, 20), *expresac* (14), *pcquosal* (20), *ovrhdr* (19, 20), *govovh* (20), *expres* (14, 16, 20, 37), *or* (19).

The following five equations describe internal interactions within the *Research Expenditures* submodel.

Similar to Eq. 23, total direct research expenses (*exprdir*) are the sum of direct research in academic units (*exprdirac*) plus direct research expenditures in the research units (*exprdirru*)

Total research-related expenditures by academic units (*expresac*) is the difference between the total research related expenditures (*expres*) and the research expenditures of the research units (*expresru*)

The following equation shows total research expenses for various research units as the direct research expenses (*exprdirru*) (defined here as all expenses for research units, since all of their output is research oriented) plus the associated indirect expenses. The overhead rate (*ovrhdrru*) is computed as the proportion of indirect costs recovered over total expenses (not just direct expenses).

Overhead amount (in dollars) of research expenditures (*ovrhdramt*) is equal to difference between total research expenditures (*expres*) and direct research expenditures (*exprdir*),

Research overhead (in dollars) associated with academic units (*ovrhdramtac*) is equal to the difference between total research overhead (*ovrhdramt*) and research overhead attributed to research units (*exprdirru*),

The remaining four equations describe interactions between the *Research Expenditures* submodel with other submodels.

The following equation calculates the direct expenditure for research in academic units (*exprdirac*). Direct academic research expenses are equal to the proportion of regular faculty academic year salaries charged to research (\(1-{ tchfrac}\)), plus the salary expense for research faculty, plus the direct expense for summer research. The last term assumes that all faculty activity during the summer is research oriented. The component \(\left( { facpcsub}+{ facpcact}\right) \times {{ mo}\over 9}\) computes the research salary amounts for the summer term.

With *Teaching Operations* and *Teaching Expenditures* submodels:

The following equation calculates an overhead rate for research (*ovrhdr*), which is based on research faculty (doing full time research) (*facres*) and the fraction of regular faculty allocated to research (\(1 - { tchfrac}\)). Note that research assistants are not included in this calculation.

With *Teaching Operations*, *Teaching Expenditures*, and *Income* submodels:

Total research expenditures are equal to direct research expenditures (see Eq. 13) times their benefit expense (*frng*), plus the total active contract expenditures for regular faculty, plus the portion of indirect expense that is actually spent but not recovered (due to federal constraints/regulations), plus the direct and indirect expenses of the research units,

Equation 21 calculates the non-research funded expenses for graduate research assistants as the total salary expense for all research assistants minus the portion of salaries charged to sponsored projects. The equation assumes $1000 per month stipend for 12 months.

### 1.4 Income Submodel

The *Income Submodel* contains equations that determine income, including distinction between “direct” and “indirect” income. The variables included in this model are: *quores* (28), *quoten* (28, 20), *facpcact* (28, 18, 20), *incresac* (21, 23, 28), *incresru* (23), *incres* (20, 23, 24, 25, 27, 31, 35, 36), *indrat* (25, 26), *resdir* (25, 26), *tuitu* (22), *aid* (22), *inctuit* (22, 24), *incmisc* (24), *incgift* (24), *incend* (24), *inc* (24, 38).

Income from tuition (*inctuit*) is the sum of the tuition income from graduate students (\({ studg} \times { tuitg}\), total number graduate students times graduate tuition) and the tuition income from undergraduate students (\({ studu} \times { tuitu}\), total number of undergraduate students times undergraduate tuition) the latter reduced by the fraction of tuition income that is devoted to financial aid (*aid*),

Research income is the sum of research-related income from academic units (*incresac*) and all income from the research units (*incresru*),

Total income is the sum of five income categories: tuition income (*inctuit*), draw on endowment (*incend*), research income (*incres*), gift income (*incgift*), and miscellenous income (*incmisc*),

Research income comes from direct (*resdir*) and indirect income. The latter is expressed as indirect research ratio (*indrat*),

Another relationship between direct and indirect research income is expressed by the following equation:

The following equation computes a dollar amount associated with the other (e.g., Faculty Administration Allowance, Sponsored Projects Administration, Library, Student Services) portion of indirect costs. The sum of *costspace*, *costda*, *costga*, and *costother* should equal total indirect costs associated with research.

Research income from academic units (*incresac*) is the sum of the average dollar amounts of active research contracts per type of researcher times the total number of researchers of that type,

### 1.5 Space Cost Submodel

The *Space Cost Submodel* contains equations describing space costs. The variables included in this model are: *rlspace* (31), *costspace* (31, 32, 33, 34), *rlda* (35), *costda* (35), *rlga* (36), *costga* (36), *rlother* (27), *costother* (27), *edfactor* (33), *bdfactor* (32), *omfactor* (34), *spaceed* (33), *spacebd* (32, 38), *spaceom* (34), *spaceru* (29), *spaceres* (29), *spacetch* (29), *space* (29, 30), *spchge* (30), *leased* (30), *capspace* (30).

The following equation assigns manpower usage of space (Note: it is not tied to expenditures.)

The followingh equation calculates a “cost of space” charge (Note, it is not tied to expenditures.)

The cost of space (in dollars) is set up as a factor of total research income. This variable is tied back to surplus (*sur*), but still needs to be tied to manpower. Equations 29 and 30 should be expanded and related to Eqs. 31, 32, 33 and 34 . One possibility would be to create four space components: classroom space as a function of students, teaching space as a function of regular and special faculty, research space as a function of research faculty, and other space as a function of overhead / total staff,

There are three components of Costspace. Equations 32, 33, and 34 compute the dollar amounts associated with each component. Equation 32 computes the portion of space costs associated with building depreciation.

Equation 33 computes the portion of space costs associated with equipment depreciation.

Equation 34 computes the portion of space costs associated with operations and maintenance of plant.

Equation 35 computes a dollar amount associated with the Departmental Administration portion of indirect costs. Interactions with *Research Expenditures* submodel.

Equation 36 computes a dollar amount associated with the General and Administrative portion of indirect costs.

### 1.6 Total Expenditures Submodel

Only one variable: *exptot* (37, 38).

Total expenditures are the sum of instructional expenditures (*exptch*), research expenditures (*expres*), and expenditures for doctoral students (*expdoc*),

### 1.7 Surplus Submodel

Only two variables: *captran* (38), *sur* (38).

Budget surplus (*sur*) is equal to the difference between total income (*inc*) and total expenditures (*exptot*) and transfers to capital (*captran*). *spacebd* is added to surplus.

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Druzdzel, M.J., Kalagnanam, J.R. Performance Budget Planning: The Case of a Research University.
*Comput Econ* **57**, 869–887 (2021). https://doi.org/10.1007/s10614-018-9799-4

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DOI: https://doi.org/10.1007/s10614-018-9799-4