Abstract
We derive and empirically apply an input-oriented distance function based on the stochastic ray production function suggested by Löthgren (1997, 2000). We show that the derived ray-based input distance function is suitable for modeling production technologies based on logarithmic functional forms (e.g., Cobb-Douglas and Translog) when control over inputs is greater than control over outputs and when some productive entities do not produce the entire set of outputs — two situations that are jointly present in various economic sectors. We also address a weakness of the stochastic ray function, namely its sensitivity to the outputs’ ordering, by using a model-selection approach and a model-averaging approach. We estimate a ray-based Translog input distance function with a data set of Danish museums. These museums have more control over their inputs than over their outputs, and many of them do not produce the entire set of outputs that is considered in our analysis. Given the importance of monotonicity conditions in efficiency analysis, we demonstrate how to impose monotonicity on ray-based input distance functions. As part of the empirical analysis, we estimate technical efficiencies, distance elasticities of the inputs and outputs, and scale elasticities and establish how the production frontier is affected by some environmental variables that are of interest to the museum sector.
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Notes
See Kumbhakar et al. (2015) regarding the importance of choosing the correct orientation for the distance function.
It would be interesting to compare results obtained with our suggested ray-based input distance functions with results obtained with corresponding traditional logarithm-based input distance functions that replace zero output quantities by arbitrarily small numbers or add small numbers to all observations of output variables that include zero values. However, we consider this to be beyond the scope of this article. Furthermore, we expect that the results would be very similar to those obtained by Henningsen et al. (2015) who compare the same functional specifications although for output-oriented distance functions instead of for input-oriented distance functions.
Modeling multiple-output firms that produce different subsets of outputs with the same distance function implicitly assumes that all firms have access to the same general production technology, e.g., that all firms are capable of producing the entire set of outputs that is considered in the analysis. Furthermore, when using functional forms that are continuous at zero output quantities, we need to assume that the analyzed production technology fulfills this assumption, i.e., \({\lim }_{{y}_{j}\to {0}^{+}}D(x,{y}_{1},\ldots ,{y}_{M},z)=D(x,{y}_{1},\ldots ,{y}_{j-1},0,{y}_{j+1},\ldots ,{y}_{M},z)\,\forall \,j\), where the mathematical symbols used here are defined in the following part of this section. The assumption of using the same general production technology is likely (approximately) fulfilled in our empirical application because all museums in our data set are likely capable of producing the entire set of outputs. Indeed, the “Consolidated Act on Museums” that regulates the museums in our data set states that these museums are expected to preserve, research and exhibit their collections. However, this assumption might not be fulfilled in other empirical applications and, thus, the suitability of this assumption needs to be carefully discussed for each empirical application of our suggested method.
The specification of the stochastic ray production frontier specified in equation (1) slightly deviates from the original specification proposed by Löthgren (1997, 2000). While the original specification takes logarithms of the angles φi, several studies that apply the stochastic ray production frontier do not take logarithms of the angles φi. As not taking logarithms of the angles usually has several advantages (Henningsen et al. 2017), we do not take logarithms of the angles φi.
We leave the derivations for other ‘directions’ of the ‘general’ distance function for future research.
A formal proof that our ray-based input distance function is indeed a Shephard input distance function is presented in Section 2 of the Online Supplement.
The Cobb-Douglas functional form is a special case with αij = 0 ∀ i, j = 1, …, N-1, βij = 0 ∀ i, j = 1, …M, δij = 0 ∀ i, j = 1, …, K, ψij = 0 ∀ i = 1, …, N-1; j = 1, …, M, ξij = 0 ∀ i = 1, …, N-1; j = 1, …, K, and ζij = 0 ∀ i = 1, …, M; j = 1, …, K. As mentioned in Section 2, we test the Translog specification defined in equation (6) against the corresponding Cobb-Douglas specification. The test rejects the Cobb-Douglas specification in favor of the Translog specification. Therefore, the following derivations are based on the Translog specification.
In Section 1 of the Online Supplement, we present the derivation of equation (6).
If an output quantity yi is zero, the distance elasticity of that output is —as expected and can be seen from equation (8)— also zero. In order to assess the effect of increasing an output quantity from zero on the distance measure, one can instead calculate semi-elasticities \(\partial \ln {D}^{i}(x,y,z)/\partial {y}_{i}\) with the right-hand side of equation (14).
If zi is in logarithm, equation (11) indicates elasticities rather than semi-elasticities.
As the method proposed by Henningsen and Henning (2009) requires monotonicity conditions that are linear in the estimated coefficients, we cannot use the monotonicity conditions ∂Di(x, y, z)/∂xi = (Di(x, y, z)/xi)(∂Di(x, y, z)/∂xi) ≥ 0 ∀ i = 1, …, N and ∂Di(x, y, z)/∂yi(Di(x, y, z)/yi)(∂Di(x, y, z)/∂yi) ≤ 0 ∀ i = 1, …, M, because they are non-linear in the estimated coefficients (given that Di(x, y, z) depends on the estimated coefficients).
State-recognized museums are not owned by the state but receive public funding and are obliged to preserve, register, research and exhibit their collections. Complying with this wide set of responsibilities entails delivering multiple cultural and educational services so that evaluating their technical efficiency in a multi-output setting is warranted.
Data for 2013 is not available.
Information about the capital stock of museums is generally not available. This has been pointed out in the existing literature for other countries (Feldstein 1991, Frey and Meier 2006, Grampp 1989, O’Hagan 1998, Peacock and Godfrey 1974) and it is also the case in Denmark. To cope with this problem, Bishop and Brand (2003) use the same proxy as we use, whereas del Barrio et al. (2009) and del Barrio-Tellado and Herrero-Prieto (2019) use the area of museums in square meters. We chose the first of these proxies because information about the area of the museums is not available in our data set.
It would be best to use an explicit measure of the output generated by conservation activities in our empirical analysis. However, as this information is unavailable, we use expenditure on conservation activities as a proxy for the output of the museums’ conservation activities.
The purpose of using the natural logarithm is to reduce the occurrence of extreme values and to reduce the skewness of the distribution of this variable.
Although ignoring the panel structure of the data set likely results in a violation of the i.i.d. assumption, we conduct a ‘pooled’ estimation because, in our data set, the variation of the variables between years is very low and the number of time periods in the data set is so small that individual effects at the museum level capture almost all variation in the variables, which means there is not enough variation to estimate the parameters of the distance function. Furthermore, we would like our efficiency scores to be based on the benchmarking of museums against each other, but a true fixed effects or a true random effects stochastic frontier model would remove the variation between the museums and, thus, prevent benchmarking between the museums. One anonymous reviewer suggested to estimate the model separately for each year. However, unfortunately, this is infeasible for our data set due to insufficient degrees of freedom as we have only 82–91 observations per year, while our model specification has 67 parameters to be estimated (see Table A1 in Section 6 of the Online Supplement).
As the ordering of the last two outputs should, theoretically, not affect the results (Henningsen et al. 2017), it should be sufficient (and computationally more efficient) to estimate the model with 360 different orderings of the outputs. As numerical inaccuracies could result in different estimates for the two models that only differ in terms of the last two outputs, we estimate the model with all 720 different orderings of the outputs in order to avoid to arbitrarily remove one ordering of each of the 360 pairs of orderings that only differ in the order of the last two outputs. However, our estimates for models that only differ in the ordering of the last two outputs are always very close to each other or are even virtually identical. Hence, estimating the models with all 720 orderings of outputs or with only 360 orderings of outputs gives virtually identical results.
In none of the 720 orderings of outputs are more than two of the last outputs zero. When the last two outputs are ‘exhibitions’ and ‘research’, there are 7 observations for which the last two outputs are zero. When the last two outputs are ‘conservation’ and ‘research’, there are 5 observations for which the last two outputs are zero. When the last two outputs are ‘conservation’ and ‘exhibitions’, there are 2 observations for which the last two outputs are zero. In all other orderings, we do not need to apply the trick.
It is worth pointing out that in our approach, the values of φi(y) and, thus, the estimation results do not depend on the size of the arbitrarily small value of κ. Hence, this is not a naive solution such as the one that replaces zero values with an arbitrarily small number.
As the number of estimated coefficients is the same for all orderings of the outputs, selecting the ordering of the outputs that gives the highest log-likelihood value is equivalent to using the Akaike information criterion (AIC) or the Bayesian information criterion (BIC) to select the ordering that gives the best fit to our data.
The interpretation of distance elasticities of ‘environmental’ variables is derived and explained in Section 5 of the Online Supplement.
Time-invariant unobserved variables could be controlled for by estimating panel data models but—as previously mentioned in footnote 17—our data set does not allow us to do this.
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Acknowledgements
We are grateful to Mette Asmild, Peter Bogetoft, and Christopher O’Donnell as well as to two anonymous reviewers and an anonymous associate editor of the Journal of Productivity Analysis for their helpful comments on earlier versions of this article. We also appreciate the comments made by participants of the North American Productivity Workshop, NAPW 2021, and the European Workshop on Efficiency and Productivity Analysis, EWEPA 2022, particularly those of Robin Sickles and Subal Kumbhakar. Of course, the authors take full responsibility for any remaining errors. We thank Lucas Alexander Kock and Berit Fruelund Kjærside from the Danish Ministry of Culture and Monika Bille Nielsen from Statistics Denmark for providing the data. Juan José Price acknowledges financial support from Copenhagen Business School (CBS) and Macquarie University.
Author contributions
JJP initiated the research, developed the general idea for the empirical analysis, collected the data, built the data set, and conducted most of the literature review. AH suggested to derive a ray-based input distance function in order to deal with zero-valued outputs in the empirical analysis. The mathematical derivations and the coding of the derived model specification was done by JJP with guidance and help from AH. JJP drafted the first version of the article, while AH revised some of its parts. Both authors participated in the revisions of the article.
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Price, J.J., Henningsen, A. A ray-based input distance function to model zero-valued output quantities: Derivation and an empirical application. J Prod Anal 60, 179–188 (2023). https://doi.org/10.1007/s11123-023-00684-1
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DOI: https://doi.org/10.1007/s11123-023-00684-1
Keywords
- Stochastic ray production frontier
- Distance function
- Input-oriented efficiency
- Zero output quantities
- Model averaging
- Museums