## Abstract

The current study examines costs, measured by educational and general (E&G) spending, and cost efficiency at 252 public master’s institutions in the United States over a nine-year (2004–2012) period. We use a multi-product quadratic cost function and results from a random-effects model with a first-order autoregressive (AR1) disturbance term to calculate economies of scale with regard to undergraduate enrollment, graduate enrollment, and research. We also employ a slightly modified version of Kumbhakar et al. (J Prod Anal 41(2):321–337, 2014) multi-step approach, involving the use of stochastic frontier analysis (SFA) and taking into account spatial interdependency, to decompose cost efficiency into long-term stable (persistent) and short-term (residual) efficiency. The key results of this study include evidence that: (1) regional clustering of costs exists; (2) there are economies of scale in undergraduate education and diseconomies of scale in graduate education; (3) relatively few institutions are cost inefficient; and (4) cost inefficiency tends to be long-term and persistent rather than short-term and residual. This research also identifies public master’s institutions that are the most cost efficient. Our inquiry has implications for future research as it points towards specific institutions, which may be engaged in effective practices to keep costs low, for possible follow-up case studies. Going forward, the techniques used in this study could be applied to examine economies of scale and scope as well as cost efficiency among other types of higher education institutions, such as public or private research universities, baccalaureate institutions, and community colleges.

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## Notes

- 1.
This was computed using IPEDS data (which includes latitudinal and longitudinal coordinates) and the Stata “nearstat” routine.

- 2.
Cheslock and associates (2016) identified other but less commonly used theoretical frameworks and perspectives that include cost disease, the positional arms race, and the principal-agent model. Cost disease is utilized to explain why costs rise faster in higher education, compared to other industries. The positional arms race perspective seeks to explain why costs increase over time within certain higher education institutions that are competing for top students and faculty. In the principal-agent model, costs are posited to be a result of the extent to which the interests of the principals (i.e., students, parents, alumni, donors, and taxpayers) diverge from those of the agents (i.e., faculty members, staff, administrators, and trustees) and tend to focus on costs in specific areas (e.g., administrative) versus total costs.

- 3.
Each higher education institution should only offer one type of output (undergraduate education, graduate education, or research).

- 4.
One higher education institution is producing all outputs.

- 5.
DEA is a nonparametric technique, based on linear programming methods to derive a non-parametric piece-wise surface (or frontier). DEA methods do not require distributional assumptions regarding deviations from the frontier. However, DEA has several limitations that include measurement error and outliers that may influence the shape and position of the frontier. For an introduction to DEA, see Thanassoulis (2001).

- 6.
When estimating a SFA model, it is assumed that the distribution of the inefficiency component of the error (

*u*_{ i }) is usually half normal, truncated normal, exponential, or normal gamma. In this study, the SFA models were estimated assuming a half-normal of*u*_{ i }. SFA models with the truncated distribution did not converge. In this research, efficiency scores generated from SFA models with an exponential distribution are highly correlated (e.g.,*r*= 0.97,*p*< 0.001) with efficiency scores generated from SFA models with a half-normal of*u*_{ i }and, consequently, were not included in the analyses. The gamma distribution, an extension of the exponential distribution of*u*_{ i }, has not been used extensively in the SFA literature and requires a complex or simulated log-likelihood function. Therefore, the gamma distribution is not used in this study. - 7.
The

*v*is assumed to be independent and identically distributed as*N*(0,*δ*^{2}), independent of the*u*, which is an unobservable nonnegative random variable assumed to be independent and identically distributed (i.i.d) as a truncation at zero of*N*(*µ, δ*^{2}). As in OLS regression models, the independent variables are assumed to be independent of the random error (*v*). - 8.
This method is based on the following formula:

\(E[\exp (u_{i} )\left| {e_{i} } \right.] = \frac{{1 - \varPhi \left( {\delta + \left( {\frac{{\gamma e_{i} }}{\delta }} \right)} \right)}}{{1 - \varPhi \left( {\frac{{\gamma e_{i} }}{\delta }} \right)}} \cdot { \exp }\left( {\gamma e_{i} + \left( {\frac{{\delta^{2} }}{2}} \right)} \right)\)

where Φ is the cumulative density function, δ = δ

_{u}. δ_{u}/δ, γ = δ^{2}_{u}/δ^{2}, and*e*_{ i }=*v*_{ i }*− u*_{ i }. - 9.
A detail description of this three-step analysis can be found in the book by Kumbhakar et al. (2015). A

*practitioner's guide to stochastic frontier analysis using Stata.*New York, NY: Cambridge University Press.We modified the Stata code provided by Kumbhakar et al. (2015) at https://sites.google.com/site/sfbook2014/to apply to our study.

- 10.
In our study, we used a maximum distance band that would also include public master’s institutions in the US, including Alaska but excluding Hawaii.

- 11.
As of January 1st, 2015, 29 states and the District of Columbia have minimum wage laws well above the federal level, thus implicitly pushing all other wages in the respective states upward. See http://www.dol.gov/whd/minwage/america.htm.

- 12.
Following the literature, we first define the

*average incremental cost (AIC)*for undergraduate output (FTE undergraduate students) as AIC_{U}= [TC{Q_{U},Q_{G},Q_{R}} − TC{0,Q_{G},Q_{R}}]/Q_{U}, where TC{Q_{U},Q_{G},Q_{R}} is the total cost of producing Q_{U}units of undergraduate students, Q_{G}units of graduate students and Q_{R}units of research; and TC{0,Q_{G},Q_{R}} is the total cost when output for Q_{U}(FTE undergraduate enrollment) is zero. We then define average incremental costs for graduate and research outputs in a similar manner. Next, we calculate the*marginal cost*(MC) of increasing the undergraduate FTE enrollment by one student by simply taking the first order derivative of the total cost function with respect to the undergraduate FTE enrollment: MC_{U}= ∂TC/∂Q_{U}. We then define and calculate marginal costs for graduate FTE enrollment and research outputs in a similar manner.*Economies of scale*are said to exist for a product when the ratio of these two measures (AIC_{i}/MC_{i}) is greater than one. If the ratio of AIC_{i}/MC_{i}is less than one,*dis*e*conomies of scale*exist. - 13.
Although this was not a part of our study, we allowed for heterogeneity in the cost inefficiency term and estimated the determinants of residual (i.e., short-term) and persistent (i.e., long-term) cost inefficiency. More specifically, we explored the relationship between (1) short-term cost inefficiency and some of the variables in our AR1 random-effects model and (2) long-term cost inefficiency and several of the variables in our AR1 random-effects model. Preliminary results showed that short-term cost inefficiency is negatively influenced by faculty salary (beta = −4.19,

*p*< 0.001), positively associated with student-to-faculty ratio (beta = 0.84,*p*< 0.01) and positively related to offering a doctoral degree (beta = 1.96,*p*< 0.001). Persistent cost inefficiency is positively related to the student-to-faculty ratio (beta = 7.80,*p*< 0.001) and the presence of an affiliated medical school (beta = 3.64,*p*< 0.01) and negatively related to faculty salaries (beta = −4.46,*p*< 0.001) and the university offering doctoral degrees (beta = −2.46,*p*< 0.001). In future research, we plan to further pursue this particular line of inquiry in a more comprehensive and systematic way. - 14.
The complete lists of all 252 institutions ranked by persistent cost efficiency scores generated by all four models are available from the authors upon request.

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## Acknowledgments

This study acknowledges Ozan Jaquette (University of Arizona), who provided access to the dataset used in this research, Kevin Eagan (UCLA), who helped to extract key variables, Manuel S. Gonzalez Canche (University of Georgia), who provided code and assistance to help create spatial maps, and the suggestions of two anonymous reviewers. Any errors are our own.

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Titus, M.A., Vamosiu, A. & McClure, K.R. Are Public Master’s Institutions Cost Efficient? A Stochastic Frontier and Spatial Analysis.
*Res High Educ* **58, **469–496 (2017). https://doi.org/10.1007/s11162-016-9434-y

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### Keywords

- Public master’s institutions
- Cost efficiency
- Stochastic frontier analysis
- Spatial analysis