Multilevel flexible specification of the production function in health economics

Previous studies on hospitals’ efficiency often refer to quite restrictive functional forms for the technology (Aigner et al., 1977, J. Econom., 6, 21–37) . In this paper, referring to a study about some hospitals in Lombardy, we formulate convenient correctives to a statistical model based on the translogarithmic function—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 hierarchical structure of the data (as in Gori et al., 2002, Stat. Appl., 14, 247–275), we propose a multilevel model, ignoring for the moment the one-side error specification, typical of stochastic frontier analysis (Aigner et al., 1977, J. Econom., 6, 21–37). Given this simplification, however, we are easily able to take into account some typical econometric problems as, e.g. heteroscedasticity. The estimated production function can be used to identify the technical inefficiency of hospitals (as already seen in previous works), but also to draw some economic considerations about scale elasticity, scale efficiency and optimal resource allocation of the productive units. We will show, in fact, that for the translogarithmic specification it is possible to obtain the elasticity of the output (regarding an input) at hospital level as a weighted sum of elasticities at ward level. Analogous results can be achieved for scale elasticity, which measures how output changes in response to simultaneous inputs variation. In addition, referring to scale efficiency and to optimal resource allocation, we will consider the results of Ray (1998, J. Prod. Anal., 11, 183–194) to our context. The interpretation of the results is surely an interesting administrative instrument for decision makers in order to analyse the productive conditions of each hospital and its single wards and also to decide the preferable interventions.