COSTS, EFFICIENCY AND ECONOMIES OF SCALE AND SCOPE IN THE ENGLISH HIGHER EDUCATION SECTOR
Department of Economics Lancaster University LA1 4YX Email: G.Johnes@lancs.ac.uk
Tel: +44 1524 594215
Huddersfield University Business School University of Huddersfield Queensgate Huddersfield HD1 3DH Email: J.Johnes@hud.ac.uk Tel: +44 1484 472231
An understanding of the production and cost technology of higher education institutions is of considerable policy interest as it motivates the structure of the sector – how large universities should be, and what mix of outputs they should produce. We review the literature, and, using data for English institutions in 2013-14, apply appropriate frontier methods to model the structure of costs in this diverse sector. In doing so, we uncover information about the returns to scale and scope within the higher education sector: in particular, the class of institutions comprising larger research intensive universities and small specialist institutions could benefit from further concentrating postgraduate and research activity. We find that the universities comprising the English higher education sector are largely efficient (measured relative to observed practices) and that there is little scope for gains in technical efficiency from allocating resources on the basis of efficiency scores.
JEL Classification: Keywords:
I23 costs, efficiency, higher education
The authors acknowledge helpful comments from Alex Teytelboym, Ken Mayhew, Christopher Bliss, an anonymous referee, and participants at the following conferences: INFORMS, Philadelphia 1st to 4th November 2015; LEER Workshop, Leuven 19th to 20th November 2015; the Annual Conference of the Association for Education Finance and Policy, Denver 17th to 19th March 2016.
Governments around the world provide subsidies to higher education and so have a legitimate interest in the efficiency with which institutions convert inputs into outputs. This is no less the case in countries, such as England, where students themselves make large contributions to the cost of their education. Under the current funding mechanism for higher education in England, many students will not pay off the whole of the debt that they accrue while studying. The Resource Accounting and Budgeting (RAB) cost – the proportion of the value of student loans that, owing to the write-off of debt after 30 years, will not be repaid – is currently estimated to be between 20% to 25%1. Therefore the tuition fee charged by colleges and universities is not necessarily the same as the amount paid by students. This gives rise, potentially at least, to a market failure in which competitive pressures fail to incentivise providers to become more efficient.
Extensive work has been undertaken on evaluating efficiency in the higher education sectors of various countries. Recent examples include the United States (Agasisti and Johnes 2015), Germany (Johnes and Schwarzenberger 2011), Italy (Agasisti and Johnes 2010), and Spain (Johnes and Salas Velasco 2007). Work on the United Kingdom (UK) is of particular relevance here (see, for example, Johnes 1990; Johnes and Taylor 1990; Johnes, J 1996; Johnes et al. 2005; Johnes 2008; Thanassoulis et al. 2011; Johnes and Johnes 2013). Much of the literature on efficiency measurement has emphasised the statistical evaluation of costs (Cohn et al. 1989), since efficiency concerns how a given output can be produced at as low a cost as possible. Statistical and econometric techniques, notably those based on stochastic frontier analysis, have been developed which allow efficiency to be evaluated for each institution and hence comparisons to be made across the sector. These statistical methods do not drill down into the detail of how institutions do what they do2; rather they offer the analyst both an understanding of how costs are determined in higher education institutions (HEIs) as a whole, and a measure of the extent to which different institutions manage to produce their outputs of teaching and research efficiently. The methods provide, at a higher level of abstraction, much the same information into benchmarking exercises as do more detailed qualitative exercises, but offer the advantage of a clear focus on the front-end activities of teaching and research. A number of studies have adopted this general approach for UK higher education3 (Glass et al. 1995a; 1995b; Johnes, G 1996; Johnes 1997; 1998; Izadi et al. 2002; Johnes et al. 2005; Stevens 2005; Johnes et al. 2008b; Johnes and Johnes 2009; Thanassoulis et al. 2011).
The purpose of this paper is to undertake an empirical study of costs and efficiency in English higher education using data on the most recent available year, namely 2013-14. The paper is in 5 sections. A review of empirical studies of costs in UK higher education is presented in section 2. Section 3 discusses the methodological issues associated with estimating cost functions, and examines how
1 http://wonkhe.com/the-wire/rab-charge-official-estimate-falls-to-20-25/. 2 Unlike, for example, the Transparent Approach to Costing (TRAC) adopted by the Higher Education Funding Council for England, which measures costs based on time diaries kept by all academics. 3 Note that there are also notable studies of cost structures of higher education systems of other countries such as Japan, Italy, Spain, Portugal, the USA and Germany, respectively (Hashimoto and Cohn 1997; Agasisti and Salerno 2007; Johnes and Salas Velasco 2007; Johnes et al. 2008a; Agasisti and Johnes 2010; Johnes and Schwarzenberger 2011).
estimates of efficiency can be derived from the cost function. The results of the empirical analysis are presented in section 4. Conclusions are drawn in section 5.
2. Review of the literature on costs in higher education in the UK
As in many developed countries, higher education funding in the UK poses challenges for governments under pressure to reduce public budget deficits and for HEIs which face continuous competitive pressure to do more with less. A thorough understanding of universities’ costs and economies of scale and scope is crucial in determining how universities should be organised to make the best use of their resources.
There is now a considerable literature concerning the cost structure and efficiency of systems of higher education.4 While the earliest work on university cost functions for the UK (Verry and Layard 1975; Verry and Davies 1976) acknowledges that universities are multi-product firms (producing teaching and research), both the estimation method and specification of the cost function are restrictive since they allow for only limited economies of scale and preclude altogether the possibility of economies of scope. Indeed, the complexities of the operation of multiproduct organisations identified by Baumol et al. (1982) were first recognised in the higher education context in the seminal work of Cohn et al. (1989). Subsequent studies have exploited developments in stochastic frontier estimation methods (Aigner et al. 1977; Charnes et al. 1978; 1979) in order to combine the estimation of multi-product cost functions with estimation of efficiency (in the UK context see, for example, Johnes, G 1996; Johnes 1998; Izadi et al. 2002; Johnes et al. 2005; Stevens 2005; Thanassoulis et al. 2011).
The adoption of frontier estimation techniques to estimate cost functions and efficiency leads to analysts facing a choice of methods of analysis. While all empirical cost efficiency evaluations are theoretically rooted in the work of Farrell (1957), they have generally employed one of two main methodological approaches. The non-parametric approach of data envelopment analysis (DEA) (Charnes et al. 1978; 1979) is grounded in linear programming methods, and allows, in the evaluation of a production or cost technology, input and output weights to differ across institutions. The method consequently has an advantage when applied to a sector comprising a highly diverse set of institutions as it allows universities to pursue their own specific missions without penalising their estimated efficiency. By way of contrast, the parametric approach of stochastic frontier analysis (SFA) (Aigner et al. 1977) is less flexible in its basic form, estimating a cost function with – in its simplest variant at least – identical parameters for all units in the sector, but it has the advantage of permitting statistical inference and calculation of economies of scale and scope. Both DEA and SFA are based on the idea that production and cost functions should be viewed as frontiers that describe best currently attainable performance, not as a line of best fit that describes average performance.
Stochastic frontier analyses conducted within numerous countries have typically found that, on the whole and with some exceptions, universities are operating close to the efficiency frontier. While global (ray) economies of scale are typically close to being exhausted, it is common to find unexhausted product-specific economies of scale associated with research (for example Johnes and Salas Velasco (2007) for Spain, Johnes and Schwarzenberger (2011) for Germany, Agasisti and Johnes
4 This fits within a broader literature on efficiency exemplified by Caves and Christensen (1980), and Fried et al. (2008).
(2010) for Italy, and Agasisti and Johnes (2015) for the US). This suggests that the desirability of increased concentration of research activity is quite widespread. Studies based on DEA (for example McMillan and Datta (1998) for Canada, Abbott and Doucouliagos (2003) for Australia; and Agasisti and Johnes (2009) for Italy and England) have likewise indicated that universities tend to have high levels of efficiency relative to the frontier within each country.
Institutions within the English higher education sector are highly diverse in terms of, for example, age, size, subject mix, research intensity and external engagement. There is no reason to expect that cost structures and efficiency should be the same across all these HEIs, and parametric estimation methods have therefore needed to be adapted in order to accommodate this heterogeneity.
One way of addressing this is to focus attention on particular (pre-defined) groups of universities, and to estimate a separate cost function for each one. The English higher education sector comprises institutions that have gained university status in three waves. Traditional universities, which had university status prior to 1992, offer degree programmes across the spectrum of academic subjects and have a well-developed research mission. Indeed these institutions often describe themselves as being ‘research led’. A large cohort of other institutions received university status in 1992; these institutions have a balanced portfolio offering degree programmes across a range of academic and vocational subjects, and undertake research. Finally, since 2003 a third wave of institutions, previously colleges of higher education, have been awarded university status. This last group comprises a diverse set of institutions; some are small and specialist, and many lack a strong research mission.
Early cost studies focus solely on traditional (pre-1992) universities (Glass et al. 1995a; 1995b; Johnes, G 1996; Johnes 1998) where scale economies appear to be significant and unexhausted for the typical university. Evidence on economies of scope, meanwhile, is mixed.
Later studies use data across both pre- and post-1992 HEIs to estimate cost functions (Johnes 1997; Izadi et al. 2002), and more recent work has included English universities from across all three groupings (Johnes et al. 2005; Johnes et al. 2008b; Thanassoulis et al. 2011). These studies have taken advantage of the increasing availability of appropriate data which permit the use of panel estimation methods as a means of dealing with unobserved heterogeneity. But changes over time in the funding regime of the English higher education sector mean that obtaining a panel which is comparable over time is difficult, and this has led to modelling problems in the panel context, especially for longitudinal data over a relatively long time period (Johnes and Johnes 2013). These studies find that scale economies are close to constant or decreasing for the typical university (Johnes 1997; Izadi et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009) while diseconomies of scope (defined across all output types) are a consistent finding (Johnes 1997; Izadi et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009).
There is also clear evidence that efficiency estimates vary by mission group: former colleges of higher education appear to be least efficient, followed by post-1992 and then pre-1992 HEIs (Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009; Thanassoulis et al. 2011). There is, across the full array of institutions, a considerable range in efficiency scores; for example, Thanassoulis et al. (2011) find, using SFA, that while mean efficiency is 0.75, it varies from 0.06 to 0.99. This vast range is likely a consequence of the diversity of the HEIs in the sample. Institutions at the lower end of the distribution of efficiencies tend to have characteristics (such as quality, size or specialisation),
inadequately captured in the ‘one size fits all’ specification of the cost function, that ‘explain’ their relatively high costs. The efficiency scores attached to these institutions therefore need to be treated with considerable caution.
One way of addressing the issue of diversity adopted in a number of studies is to add a set of exogenous control variables which might affect costs into the estimated cost function. Such factors include ‘quality’ of students, input prices (as reflected by geographical location dummies), real estate costs, success in strategies to widen student participation in higher education, and measures of third mission (knowledge transfer) activity. While one study finds that the proportion of students achieving first and upper second class degrees has a positive influence on both costs and on efficiency (Stevens 2005), student quality has generally not been found to be a significant determinant of costs (Verry and Davies 1976; Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009). Accounting for third mission activity has proved to be very difficult in practice because of the paucity of data. Variables used to reflect third mission include income from other services rendered (Johnes et al. 2005), income from intellectual property (including contract research, consultancy and income from engagement with business and the community), and staff time directed at and attendees at public events (Johnes and Johnes 2013). Data around public events seem particularly unreliable; income from intellectual property has the expected positive relationship with costs (Johnes and Johnes 2013). More generally, identifying control variables that might influence costs, over and above the standard outputs of teaching and research, has not proved particularly successful in studies to date.
A small number of recent cost studies has therefore experimented with random parameter stochastic frontier models (Tsionas 2002; Greene 2005). These models do not require data on timeinvariant influences on costs, but – as an advance on fixed effects models – allow all parameters of the cost equation to vary across institutions. Much like DEA, the random parameter stochastic frontier estimation method thus effectively estimates a separate technology for each unit of observation. Applications of this method are limited with only one applied to higher education in England (Johnes and Johnes 2009) and the others to universities in Italy, Germany and the USA (Agasisti and Johnes 2010; Johnes and Schwarzenberger 2011; Agasisti and Johnes 2015). A common result of these studies is that, for the typical university, economies of scale are generally exhausted (although this is not the case for typical universities in Germany) and opportunities for savings arising from global economies of scope are limited. The disadvantage of this approach is that the model is very demanding of the data and can, in consequence, lead to a failure of the algorithms used in estimation to converge. In addition, by allowing each HEI to have its own mission and be judged in isolation, the random parameters approach might be considered to be too tolerant of high-cost practices.
An alternative to random parameter frontier estimation involves estimating separate frontiers for two or more groups of institutions. This is less permissive than the random parameters method in that it allows variation in the parameters between institutions in different classes, but no variation within each class. The latent class approach is particularly attractive because it allows the membership of each class to be determined by the data without need for the analyst to prescribe
which institutions belong in which class.5 Unlike the random parameter approach, the latent class model can be estimated using a cross-section of data; in a policy environment that is rapidly changing, this is an important advantage. To date this method has not received much attention in the higher education literature (exceptions include Johnes and Johnes 2013; Agasisti and Johnes 2015). This is a gap which we intend to fill with this study.
3. Methodological approach
We now describe our approach to the key issues in undertaking an analysis of university costs and efficiency mentioned above: the functional form of the cost function, the modelling of economies of scale and scope, and the choice of estimation technique.
3.1 Functional form
A cost function relates costs to the set of outputs produced, given prices of inputs. For institution ???????? this is written in the general form
???????????????? = ????????(????????????????????????; ????????????????????????)
where ???????????????? represents costs for university ????????, ???????????????????????? is quantity of output ???????? for university ???????? and ???????????????????????? is the price of input ???????? for university ????????. We estimate a quadratic cost function; this satisfies the desiderata identified by Baumol et al. (1982, pp448-450) – including the requirement that the equation should produce sensible estimates of costs when there is zero production of one or more of the outputs. The specification is therefore:
???????????????????????? ???????????????????????? ????????????????????????
where ???????????????? is an institution-specific residual, and ????????0, ????????????????, ???????????????????????? and ???????????????? are parameters to be estimated.
3.2 Economics of scale and scope
Measures of the returns to scale and scope suggested by the estimated cost function are evaluated following Baumol et al. (1982). These measures are defined in Table 1. In the case of returns to scale, the measures all draw on the idea, familiar from the literature on single-product firms, that average costs are higher than marginal costs over the range of output where the former are decreasing. Where the measure of ray or product-specific returns to scale exceeds unity, there are increasing returns to scale; where the measure is below unity, returns to scale are decreasing. The evaluation of global economies of scope involves examining the cost of producing all the outputs of the typical university together and comparing that to the sum of the costs of producing each output (at the same level) in separate production units. Product-specific economies of scope refer to the cost savings (or otherwise) of producing one specific output along with all the others. Economies of scope (global or product-specific) are observed when the corresponding measure is positive.
5 To balance against this advantage, a possible disadvantage of the method is that it is not always possible to identify the model; it may not be possible to allocated observations into classes with a high degree of confidence.
3.3 Estimation method
We have identified problems of estimating higher education cost functions for a diverse higher education sector such as that observed in England. Panel data estimation with a random parameters specification might offer a way forward, but recent changes in the student funding mechanism – notably the increase in undergraduate tuition fees to £9000 – limit the extent to which data from different years are comparable. In this paper, we therefore apply a latent class stochastic frontier estimation approach to cross-section data. Specifically the latent class stochastic frontier model for each class ???????? is
????????????????????????,???????? ???????????????????????? ????????????????????????
???????????????? ,???????? ????????????????????????
Equation (3) is estimated, with the analyst prescribing how many latent classes exist, but with the membership of each latent class being determined alongside the parameter values and the residual terms by maximum likelihood.6 The ???????? and ???????? terms represent respectively a normal residual and an asymmetric residual. The latter, which in the present paper we assume to follow a half-normal distribution, is designed to capture additions to costs that are due to inefficiency. The one-sided nature of this residual means that the parameters of the cost function in (3) define a curve that, rather than providing the best fit to the data, represents an envelope around points on the efficient frontier.
To summarise, it is possible to combine the stochastic frontier and latent class approaches so that (i) cost frontiers (or envelopes) are estimated (ii) yielding measures of the efficiency of each organisation in the data set and (iii) establishing which organisations belong in each of the latent classes or groups. It is useful to illustrate this method graphically. Consider Figure 1. This shows a scatter plot of points, each of which describes the costs and output levels of a single observation. Each observation might represent a decision-making unit or organisation – in our case a higher education institution. A straightforward latent class analysis of these data might involve the analyst in specifying that there are two different types of institution in the data set. On the assumption of a linear functional form, the latent class model therefore fits two lines to the data. These are shown by the two dashed lines. In fitting these two lines, the model also determines which observations belong to which of the two latent classes – thus the model classifies some of the cost-output pairings into class X and some into class Y. These letters are shown as the data points on the diagram, but it should be emphasised that the observations are placed in these classes by the maximum likelihood algorithm used in the latent class estimation itself; the observations are not placed within one class or the other by the analyst.
Now the two dashed lines represent the best fit that is associated with the observations (given that there are two latent classes), but they do not represent the cost envelope faced by organisations within each of these two classes. To find these cost envelopes, the latent class method must be used alongside a stochastic frontier model. Doing this moves the lines down (though this is not necessarily
6 Methods used to help the analyst choose the appropriate number of latent classes include bootstrapping – see, for example, van der Heijden et al. (1997). In the present exercise we use the Akaike Information Criterion.
a parallel shift). The resultant cost envelopes are represented by the solid lines. Note that, within each latent class, some observations lie below the cost frontier (because of the stochastic error component), but most lie above. The preponderance of observations above the frontiers represents inefficiency. The technique allows the efficiency of each observation to be evaluated by reference to its position relative to the frontier for the latent class to which the observation belongs.7
4. Empirical analysis
Data on costs and outputs in higher education are published by the Higher Education Statistics Agency (HESA) in a series of annual publications, including Students in Higher Education and Finances of Higher Education Providers. In the analysis that follows, we draw on these data to estimate a frontier model of costs as a function of several outputs and input prices, recognising the multiproduct nature of higher education.
The costs variable includes current expenditures excluding ‘hotel’ costs associated with residences and catering. Student numbers – classified, in the case of undergraduates, into the sciences and other subjects – are expressed as full-time equivalents.8 We eschew the option to employ a finer disaggregation of the student body owing to problems of multicollinearity. Following the precedent set by earlier studies, we use research income as the measure of research activity. This measure has the virtue of providing a market value for research, hence appropriately weighting quantity and quality. While it is a prospective measure, and may be criticised for being an input rather than an output, it is typically highly correlated with measures (such as publications or citations) that are more unambiguously considered to be research outputs, but which are more retrospective in nature (Johnes and Johnes 2013).9
Previous analyses of university costs have, with few exceptions, failed to control for variations in costs due to the impact of labour market conditions differentially affecting institutions. An important exception – albeit one that predates the use of frontier methods – is that of Cohn et al. (1989) where wage is found to be a significant variable in the cost function. In this paper, we use as a control a measure of hedonic costs in the labour market derived as the residual from a regression of institutions’ salary costs against a vector of variables describing the numbers of staff in each of ten age groups. Definitions of all variables can be found in Table 2.
7 This is done using a method developed by Jondrow et al. (1982). Note that the frontier is defined – as must necessarily be the case – by reference to the data in the sample. This means that an analysis of English institutions allows efficiency to be measured only by reference to those institutions, not to an absolute standard. Analyses of the efficiency of higher education institutions across countries are scarce, but include, for example, Agasisti and Johnes (2009), Agasisti and Pérez-Esparrells (2010), Agasisti (2011), and Agasisti and Haelermans (2015). 8 In earlier work we have experimented with measures of student quality (Johnes et al. 2008b). Quality measures proved to be statistically insignificant in that work. 9 A superficially appealing alternative would be to use data from the Funding Council’s Research Excellence Framework. These data explicitly evaluate past performance in research. We note, however that these data directly determine quality related funding from the Funding Council, and so they are very highly correlated with the measure used here.
The sample used for estimation comprises 103 higher education institutions in England.10 In Table 3, the coefficient estimates produced by two alternative specifications of the model are presented. In the first column appear (for comparison purposes) the results of a straightforward stochastic frontier analysis; this assumes no unobserved heterogeneity across institutions. The remaining columns report coefficients obtained by applying a latent class stochastic frontier model (with 2 classes) to the data. In both cases, we employ a quadratic specification to capture returns to scale and scope, the latter being due to synergies across the various teaching activities and research.11
These results are not amenable to straightforward interpretation, and we defer to later a consideration of what they imply about cost structures in the higher education sector.12 First, we examine the composition of the two latent classes. Recall that the classification of institutions into one class or the other is determined (alongside the coefficient estimates and the one-sided residual that captures inefficiency) by the criterion that this should optimise the fit of the model to the data. In essence, the latent classes each contain institutions that are, in some sense not directly observed in the variables, similar to one another but are distinct from those in the other class. Using the latent class approach thus accommodates a degree (albeit limited) of unobserved heterogeneity across institutions, and a look at the composition of each class should teach us something about which institutions are alike and which are not.
Table 4 reports descriptive statistics for the key variables for the institutions belonging to each class, and details of class membership appear in Table 5. On most measures, the means of the variables are quite similar across classes, though the extent of research activity is on average somewhat higher in the second latent class than in the first. More detailed investigation shows some more pronounced differences between the two classes, however. Latent class 2 contains the largest universities, and also includes many of the smaller institutions that have gained university status since the turn of the millennium (see Table 5 for details). The standard deviation attached to all variables is correspondingly greater for this latent class than for the first.13
10 We exclude from the sample a number of institutions that are, for one reason or another, idiosyncratic. These are: the ancient universities of Oxford and Cambridge, whose costs are affected by their internal structures and teaching methods; small and specialist institutions with costs below £25m per year; the University of Arts, London, for which, owing to an unusual employment structure, we were unable to obtain hedonic salary cost; Buckingham, which is fully private; Open University, which specialises in distance learning; and the London University (Institutes and Activities), which comprises a number of highly specialised research centres. Data for University Campus, Suffolk are divided equally between the Universities of Essex and East Anglia; figures for Liverpool Tropical Medicine are added to Liverpool University. 11 The Akaike Information Criterion rejects the 1-class model (a conventional model applied across the whole sample) in favour of the 2-class latent class model at a conventional significance level. 12 Relatively few coefficients differ significantly from zero in the latent class mdoel. This is usual in this type of analysis, given the highly nonlinear nature of the functional form. 13 The mixed character of the institutions in the second latent class begs the question of whether the number of classes ought to be extended. We have tried to do this, but statistical considerations (specifically a singular variance matrix) prevent estimation when using a quadratic functional form. We report a sensitivity check of the results using a linear functional form later in the section.
In Table 6, we report the levels of average incremental costs for each output – both for the straightforward stochastic frontier and for the case in which the frontier method is applied to two latent classes. In the case of the latent class model, the calculations are provided for the institution producing, respectively, an average output vector, twice and half the average output vector, in each of the latent classes. It is worth noting that an average institution (or one that is double or half the size of the typical institution) is a hypothetical construct; in reality, HEIs may specialise to a greater or lesser extent in a particular output.
The costs associated with undergraduate education for the typical university average between about £3000 and £8000, with a greater dispersion in latent class 2 (which has a greater preponderance of both universities with medical schools and small, specialist institutions) than in latent class 1. These figures are low in comparison with tuition fees, which in most cases amount to the maximum permitted £9000 for domestic and EU students. Postgraduate tuition involves institutions in (considerably) greater expenditure per student. This is unsurprising in view of the facts that classes for taught programmes are often relatively small, and that research requires one-on-one supervision and is undertaken year round (with no long vacation). The average incremental cost of research is high, at around 2.1 – indicating that each pound of research income (the measure of research used here) is associated with more than twice as much expenditure. This confirms the conventional wisdom that income from teaching is used to cross-subsidise research. The main distinction between classes 1 and 2 is that postgraduate costs are considerably higher in class 2 than class 1. The general pattern of average costs across output types in both classes is similar for the scenarios in which output is greater than or less than the average.
We check the sensitivity of the results to both functional form and number of classes by estimating alternative models. The composition of the two groups from our quadratic latent class estimation suggests that the second class comprises two types of institution: small and large ones. In addition, English HEIs are typically categorised into three classes: pre-1992, post-1992 and former colleges of higher education. A 3-class latent class model might therefore seem appropriate in this context, and we report in the Appendix details of a 3-class model with a linear functional form14. The results are reassuring in the sense that average incremental costs (Table A1) are largely in the same range as those reported for the quadratic latent class model in Table 6. An examination of the mean values for costs and outputs (Table A2) suggests that the universities are grouped into small institutions (class 1), large research-intensive institutions (class 2) and large but less research-intensive universities (class 3).
In Table 7, we report estimates of product-specific and ray returns to scale for institutions, first for the straightforward model with no latent classes, and secondly within each latent class for the HEI of average, twice and half the average output vector. Ray economies of scale are unambiguously exhausted across both classes at large sizes, since the measure is below unity; but there are potential economies for the smallest HEIs (in both classes) and also for typical universities in class 1. Note that the typical HEI in class 2, for which ray economies of scale are already exhausted, is larger in some dimensions than its counterpart in class 1.
14 A three latent class quadratic model failed to converge.