Economies of scale and scope in the provision of diagnostic techniques and therapeutic services in Portuguese hospitals

Abstract

This article analyses the provision of auxiliary clinical services that are typically carried out within the hospital. We estimate a flexible cost function for three of the most important (cost-wise) diagnostic techniques and therapeutic services in Portuguese hospitals: Clinical Pathology, Medical Imaging, and Physical Medicine and Rehabilitation. Our objective in carrying out this estimation is the evaluation of economies of scale and scope in the provision of these services. For all services, we find evidence of economies of scale and some evidence of economies of scope. We also find evidence of diminishing returns to management, whereby larger hospitals appear to have surpassed their optimal size. These results have important policy implications and can be related to the ongoing discussion of where and how should hospitals provide these services.

Keywords:

• translog cost function
• economies of scale
• economies of scope
• clinical services
• hospitals

JEL Classification:

• D24
• I12
• I18

Acknowledgements

We would like to thank participants of the 11th Portuguese National Health Economics Conference (Porto, October 2009) and an anonymous referee for their useful comments and suggestions. We would also like to thank Sofia Nogueira da Silva for her collaboration at an early stage of this article. Financial support from Foundation for Science and Technology (FCT) and POCI 2010 is gratefully acknowledged.

Notes

¹ On efficiency, see, among others, Zuckerman et al. (1994), Rosko and Chilingerian (1999), Rosko (2001), Staat (2006) or Herr (2008). On cost structures, see, for example, Cowing and Holtmann (1983), Grannemann et al. (1986), Vitaliano (1987), Vita (1990), Fournier and Mitchell (1992), Aletras (1999), Li and Rosenman (2001) or Preyra and Pink (2006).

² Coase (1937, pp. 394–5).

³ We assume λ = 0.1. An earlier version of this article (Gonçalves and Barros, 2009) shows that our results are robust to different values of λ.

⁴ For the fixed factor, k, for each output y _i (i = 1, … , n) and for each input price w _j (j = 1, … , m) we divide each observation by the respective mean, and hence the mean of the (new) mean-scaled variables is equal to 1.

⁵ A previous version of the article (Gonçalves and Barros, 2009) estimates a second variant of each model under the assumption of homotheticity: as Smet (2002) notes, homotheticity implies that the mix of inputs which minimizes costs is not affected by the volume or even the mix of outputs and, therefore, changes in input prices will affect costs by a scale factor. In practice, homotheticity implies that δ _ri  = 0, ∀r, i, in Equation 4, i.e. input prices are not interacted with output levels. However, the homotheticity assumption is not rejected only for one model (Clinical Pathology, model 1) and we have focused instead on the results from the unrestricted (nonhomothetic) models.

⁶ The quantities produced by each hospital of the various outputs and/or diagnostic techniques and therapeutic services are also provided.

⁷ In 2004, they accounted for 56% of the total costs of diagnostic techniques and therapeutic services.

⁸ The casemix index for the years 2005 and 2006 for some ‘EPE’ hospitals was not publicly available. In those cases, and because the casemix index does not change significantly over time, we have assumed that those hospitals' casemix index was equal to that of the most recently available year.

⁹ The acronyms ‘EPE’ and ‘SPA’ stand for Entidade Pública Empresarial and Sector Público Administrativo, respectively.

¹⁰ Each output's weight in overall speciality costs varies significantly and within each speciality one output typically stands out in terms of its share of total costs. For instance, Clinical Chemistry is the output responsible for 57% of the total costs of Clinical Pathology. Similarly, Radiology accounts for some 74% of total Medical Imaging costs and Physical therapy accounts for 60% of total Physical Medicine and Rehabilitation costs. See Gonçalves and Barros (2009) for more details.

¹¹ In other words, we have not considered observations for which there was clearly misreported output production.

¹² Portuguese hospitals are divided in three hierarchical categories: central, district and level 1 hospitals; therefore, a dummy variable was created taking on the value of 1 for the latter two categories (central hospitals were omitted): ‘D–district hospital’ and ‘D–Level 1 hospital’ . There are five regions in Portugal and a dummy variable was created for four of those regions (the Alentejo region was omitted): ‘D–Region Algarve’; ‘D–Region Centro’; ‘D–Region L. V. Tejo’ (which includes Portugal's capital and largest city – Lisbon); ‘D–Region Norte’ (which includes Portugal's second largest city – Porto). Finally, dummy variables were introduced for each year (except 2002, which was omitted): ‘D–2003’; ‘D–2004’; ‘D–2005’; ‘D–2006’.

¹³ We thank an anonymous referee for suggesting this line of analysis.

¹⁴ For instance, Gujarati (1995, p. 335) suggests that correlations in excess of 0.8 may indicate collinearity, although such correlations are a necessary but not sufficient condition for its presence. The highest pair-wise correlation we have found between these variables (across specialities) was 0.81.

¹⁵ The VIFs inform us on how the variance of an estimator is inflated by the presence of multicollinearity (see Maddala (1992, p. 274) or Gujarati (1995, p. 328) for more details). VIFs above 10 are typically a sign that the variable is collinear. Looking only at these variables, none presents a VIF above five, thus suggesting no multicollinearity problems.

¹⁶ The condition number or index is an overall measure of multicollinearity and measures the sensitivity of regression estimates to small changes in the data (see Maddala (1992, p. 274) for details). The condition numbers of these variables for each speciality were below 16 (values between 10 and 30 are indicative of moderate multicollinearity and above 30 multicollinearity is severe).

¹⁷ The ad-hoc specification adopted is a mixture of a translog (costs and input prices enter the regression in logarithms) and a quadratic cost function (outputs do not enter the regression in logs because of zero output levels).

¹⁸ The standard approach of estimating the full cost function and presenting all the estimated coefficients, as in Table 2, may also be justified because the coefficients remain unbiased even in the presence of multicollinearity.

¹⁹ See Maddala (1992, pp. 209–11) for more details.

²⁰ Gujarati (1995, p. 368) suggests that such informal methods are useful to detect the possible existence of a relationship between the fitted values and the residuals, thus informing us on the type of heteroscedasticity that may be present.

²¹ The RESET test (see Maddala, 1992, p. 204 for more details) consists of regressing û (the estimated residuals) on ŷ ² and ŷ ³, where ŷ are the fitted values, and testing whether the respective coefficients are significant (in which case the hypothesis of homoscedasticity would be rejected).

²² The hypothesis of homoscedasticity could not be rejected (for all models and specialities) at the 1% significance level. It would be rejected at the 5% significance level for Physical Medicine and Rehabilitation (model 1) and at the 10% significance level for Clinical Pathology (model 1).

²³ Within each hospital, and for all specialities and models, we have looked at the correlation between the residual at time t and at time t − 1. For Clinical Pathology (81 hospitals) and Medical Imaging (83 hospitals), no correlation is significant at the 1% significance level; for Physical Medicine and Rehabilitation (74 hospitals), one correlation (in model 2) is significant at the 1% level.

²⁴ See Vita (1990) for a more detailed discussion.

²⁵ In the calculation of RTS (according to Equation 7) all the relevant coefficients are used (even statistically insignificant ones). As discussed earlier, the presence of multicollinearity causes the coefficients' SEs to be larger, but their estimates remain unbiased.

²⁶ See, for instance, Cowing and Holtmann (1983) or Vita (1990).

²⁷ Our results indicate that the mean of the individually estimated RTS estimates is heavily influenced by very high and very low RTS estimates, which are clearly related to their distance from the approximation point. Therefore, we have chosen to present the median of such estimates, which is not influenced by their magnitude.

²⁸ As mentioned earlier, multicollinearity causes SEs to be large, thus making it more difficult to find statistically significant estimates for C _vij and C _vjK .

²⁹ One could also conclude there to be suggestive evidence of WCC between outputs 1 and 7 (clinical chemistry and clinical haematology/haematoncology) and 4 and 7 (clinical microbiology and clinical haematology/haematoncology), insofar as the coefficients satisfy the more stringent criteria (but not the less stringent criteria) across models.

³⁰ One could also argue that there may be some evidence of economies of scope between outputs 3 and 4 (mammography and computed tomography), insofar as the less stringent criteria is satisfied for both models.

³¹ And this may imply higher maintenance costs.

³² We omit the results from the article but we are happy to provide them upon request.

³³ Coase (1937, pp. 394–5).

³⁴ As in Table 2, except for Physical Medicine and Rehabilitation.

³⁵ The hypothesis of constant returns to scale is clearly rejected for Clinical Pathology and Medical Imaging. However, we must beware that the underlying estimates may be biased because of the specification bias.