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Measuring performance in the presence of stochastic demand for hospital services: an analysis of Belgian general care hospitals


Since demand for hospital services is subject to substantial variability, the relationship between uncertain demand, excess capacity, hospital costs and performance should be investigated thoroughly. In this paper a waiting time indicator to proxy hospital standby capacity is incorporated into a multi-product translog cost function for Belgian general care hospitals. The indicator is derived from queuing theory and improves on the conventionally used (inverse of the) occupancy rate. The multi-product stochastic frontier specification allows calculation of cost elasticities and marginal cost of seven hospital departments, as well as the degree of economies of scale and scope and enables identification of differences in efficiency.

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  1. If this assumption is left, calculation of the typical queuing indicators becomes very complicated and it is often not possible to derive exact closed form expressions.

  2. The typical queuing term “utilisation rate” is equal to the “occupancy rate” which was used in the sections describing hospital cost function studies that accounted for uncertain demand.

  3. Unfortunately we don’t have data on the number of beds per department.

  4. Using L’Hôpital’s rule: \({\mathop {\lim }\limits_{\lambda _B \to 0} \frac{y_i^{\lambda _B} -1}{\lambda _B }=\ln y_i }\)

  5. National Institute for Sickness and Invalidity Insurance.

  6. Under this system inefficient hospitals have their budget decreased.

  7. 82 specialised hospitals (mostly psychiatric) were not included in this analyses. Because of their nature they often only provide one specific treatment (output) which makes it almost impossible to compare them with general care hospitals.

  8. Because of the long history of involvement of the Catholic Church in the provision of health care, the majority of hospital care is provided by state-subsidised, not-for-profit (catholic) private initiative, i.e. most private hospitals are owned by religious charitable orders. The public and the private (not-for-profit) sectors operate in the same market and receive more or less comparable levels of resources.

  9. A model including input prices and using cost share equations generated poor results.

  10. It is also possible to specify own starting values.

  11. Unfortunately, more detailed variables such as a casemix indicator were not available.

  12. HHI is calculated as the sum of the squared local market shares (measured in number of beds).

  13. These indicators are on the level of the town in which the hospital is located and were obtained from ECODATA, FOD Economie, K.M.O., Middenstand &Energie.


  • Bagust A, Place M, Posnett JW (1999) Dynamics of bed use in accommodating emergency admission: stochastic simulation model. Br Med J 319:155–158

    Google Scholar 

  • Battese GE, Coelli TJ (1995) A model for technical inefficiency effects in a stochastic frontier production function for panel data. Emp Econ 20:325–332

    Article  Google Scholar 

  • Baumol WJ, Panzar JC, Willig RD (1988) Contestable markets and the theory of industry structure. Harcourt Brace Jovanovich, San Diego

    Google Scholar 

  • Bilodeau D, Crémieux P, Ouellette P (2000) Hospital cost function in a non-market health care system. Rev Econ Stat 82:489–498

    Article  Google Scholar 

  • Carey K (1998) Stochastic demand for hospitals and optimizing “excess” bed capacity. J Reg Econ 14:165–187

    Article  Google Scholar 

  • Caves DW, Christensen LR, Tretheway MW (1980) Flexible cost functions for multiproduct firms. Rev Econ Stat 62:477–481

    Article  Google Scholar 

  • Coelli T (1995) Estimators and hypothesis tests for a stochastic production frontier function: a monte carlo analysis. J Prod Anal 6:247–268

    Article  Google Scholar 

  • Coelli T (1996) A Guide to FRONTIER Version 4.1: A computer program for stochastic frontier production and cost function estimation, centre for efficiency and productivity analysis. University of New England, Armidale, Australia, Working Paper, no. 96/07, 1–33

  • Coelli T, Rao DS, Prasada Battese GE (1998) An introduction to efficiency and productivity analysis. Kluwer Academic Publishers, Boston/Dordrecht/London

    Google Scholar 

  • Conrad RF, Strauss RP (1983) A multiple-output multiple-input model of the hospital industry in North Carolina. Appl Econ 15:341–352

    Google Scholar 

  • Cowing TG, Holtmann AG (1983) Multiproduct short-run hospital cost functions: empirical evidence and policy implications from cross-section data. South Econ J 49:637–653

    Article  Google Scholar 

  • Dawson D, Goddard M, Street A (2001) Improving performance in public hospitals: a role for comparative costs? Health Policy 57:235–248

    Article  Google Scholar 

  • Folland ST, Hofler RA (2001) How reliable are hospital efficiency estimates? exploiting the dual homothetic production. Health Econ 10:683–698

    Article  Google Scholar 

  • Friedman B, Pauly M (1981) Cost functions for a service firm with variable quality and stochastic demand: the case of hospitals. Rev Econ Stat 63:620–624

    Article  Google Scholar 

  • Fujii A (2001) Determinants and probability distribution of inefficiency in the stochastic cost frontier of Japanese hospitals. Appl Econ Lett 8:807–812

    Article  Google Scholar 

  • Fujii A., Ohta M. (1999) Stochastic cost frontier and cost inefficiency of japanese hospitals: a panal data analysis. Appl Econ Lett 6:527–532

    Article  Google Scholar 

  • Gaynor M, Anderson GF (1995) Uncertain demand the structure of hospital costs and the cost of empty hospital beds. J Health Econ 14:291–317

    Article  Google Scholar 

  • Graham GG, Cowing TG (1997) Hospital reserve margins: structural determinants and policy implications using cross-section data. South Econ J 63:692–709

    Article  Google Scholar 

  • Grannemann TW, Brown RS, Pauly MV (1986) Estimating hospital costs. a multiple-output analysis. J Health Econ 5:107–127

    Article  Google Scholar 

  • Greene WH (1997) Econometric analysis, 3rd edn. Prentice Hall Upper Saddle River, New Jersey

    Google Scholar 

  • Hillier FS, Lieberman GJ (1995) Introduction to operations research, 6th edn. Industrial Engineering Series, McGraw-Hill, New York e.a.

    Google Scholar 

  • Hughes D, McGuire A (2003) Stochastic demand, production responses and hospital costs. J Health Econ 22:999–1010

    Article  Google Scholar 

  • Jacobs R (2001) Alternative methods to examine hospital efficiency : data envelopment analysis and stochastic frontier analysis. Health Care Manage Sci 4:103–115

    Article  Google Scholar 

  • Jacobzone S (1999) De Beheersing van de Uitgaven en de Verantwoordelijkheid van de Actoren in de Gezondheidssystemen, in RIZIV Jaarverslag 1999, RIZIV, Brussel, pp 106–117

  • Johnson NL, Kotz S (1970) Distributions in statistics: continuous univariate distributions-1, The Houghton Mifflin series in statistics. Houghton Mifflin Company, Boston e.a.

    Google Scholar 

  • Joskow PL (1980) The effect of competition and regulation on hospital bed supply and the reservation quality of the hospital. Bell J Econ 11:421–447

    Article  Google Scholar 

  • Keeler TE, Ying JS (1996) Hospital costs and excess bed capacity: a statistical analysis. Rev Econ Stat 78:470–481

    Article  Google Scholar 

  • Li T, Rosenman R (2001) Cost inefficiency in Washington hospitals: a stochastic frontier approach using panel data. Health Care Manage Sci 4:73–81

    Article  Google Scholar 

  • Mulligan JG (1985) The stochastic determinants of hospital-bed supply. J Health Econ 4:177–181

    Article  Google Scholar 

  • Nahmias S (1993) Production and operations analysis, 2nd edn. Irwin, Burr Ridge, Illinois; Boston, Massachusetts and Sydney, Australia

  • NHS Centre for Reviews and Dissemination (1996) Concentration and choice in the provision of hospital services. The relationship between volume and the scope of activity and hospital costs, CDR Report 8 (part II), University of York, York

  • Pauly MV, Wilson P (1986) Hospital output forecasts and the cost of empty hospital beds. Health Ser Res 21:403–428

    Google Scholar 

  • Rosko MD (2001) Cost efficiency of US hospitals: a Stochastic frontier approach. Health Econ 10:539–551

    Article  Google Scholar 

  • Smet M (2002) Cost characteristics of hospitals. Soc Sci Med 55:895–906

    Article  Google Scholar 

  • Smet M (2004) Multi-product costs and standby capacity derived from Queuing Theory: the case of Belgian hospitals. Appl Econ 36:1475–1487

    Article  Google Scholar 

  • Street A, Jacobs R (2002) Relative performance evaluation of the English acute hospital sector. Appl Econ 34:1109–1119

    Article  Google Scholar 

  • Vita MG (1990) Exploring hospital production relationships with flexible functional forms. J Health Econ 9:1–21

    Article  Google Scholar 

  • Webster R, Kennedy S, Johnson L (1998) Comparing techniques for measuring the efficiency and productivity of Australian private hospitals, Working papers in econometrics and applied statistics. Australian Bureau of Statistics, no. No. 98/3, 1–60

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The author would like to thank Diana De Graeve, Walter Nonneman, Wilfied Pauwels and two anonymous referees of this journal for their useful comments on ealier versions of this paper. I also wish to thank the Belgian Federal Ministry of Social Affairs, Public Health and the Environment for providing the data for this study. However, the author is the sole responsible for the empirical analysis and conclusions presented here.

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Correspondence to Mike Smet.

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Smet, M. Measuring performance in the presence of stochastic demand for hospital services: an analysis of Belgian general care hospitals. J Prod Anal 27, 13–29 (2007).

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  • Hospital costs
  • Stochastic demand
  • Efficiency
  • Productivity
  • Stochastic frontier analysis
  • Econometrics
  • Queuing theory
  • Multi-product cost function

JEL Classifications

  • C01
  • C13
  • C21
  • D24
  • H51
  • l11
  • l12