Multilevel flexible specification of the production function in health economics

^** Email: grassetti{at}stat.unipd.it^*** Email: e.gori{at}dss.uniud.it^**** Email: simona.minotti{at}unicatt.it Previous studies on hospitals' efficiency often refer to quiterestrictive functional forms for the technology (Aigner et al.,1977, J. Econom., 6, 21–37). In this paper, referringto a study about some hospitals in Lombardy, we formulate convenientcorrectives to a statistical model based on the translogarithmicfunction—the most widely used flexible functional form(Christensen et al., 1973, Rev. Econ. Stat., 55, 28–45).More specifically, in order to take into consideration the hierarchicalstructure of the data (as in Gori et al., 2002, Stat. Appl.,14, 247–275), we propose a multilevel model, ignoringfor the moment the one-side error specification, typical ofstochastic frontier analysis (Aigner et al., 1977, J. Econom.,6, 21–37). Given this simplification, however, we areeasily able to take into account some typical econometric problemsas, e.g. heteroscedasticity. The estimated production functioncan be used to identify the technical inefficiency of hospitals(as already seen in previous works), but also to draw some economicconsiderations about scale elasticity, scale efficiency andoptimal resource allocation of the productive units. We willshow, in fact, that for the translogarithmic specification itis possible to obtain the elasticity of the output (regardingan input) at hospital level as a weighted sum of elasticitiesat ward level. Analogous results can be achieved for scale elasticity,which measures how output changes in response to simultaneousinputs variation. In addition, referring to scale efficiencyand to optimal resource allocation, we will consider the resultsof Ray (1998, J. Prod. Anal., 11, 183–194) to our context.The interpretation of the results is surely an interesting administrativeinstrument for decision makers in order to analyse the productiveconditions of each hospital and its single wards and also todecide the preferable interventions.