Does prospective payment increase hospital (in)efficiency? Evidence from the Swiss hospital sector

Abstract

Several European countries have followed the USA in introducing prospective payment for hospitals with the expectation of achieving cost efficiency gains. This article examines whether theoretical expectations of cost efficiency gains can be empirically confirmed. In contrast to previous studies, the analysis of hospitals in Switzerland provides a comparison of a retrospective per diem payment system with a prospective global budget and a payment per patient case system. Using a sample of approximately 90 public financed Swiss hospitals during the years 2004–2009 and Bayesian inference of a standard and a random parameter frontier model, cost efficiency gains are found, particularly with payment per patient case. Prospective payment, designed to put hospitals at operating risk, is more effective in terms of cost reduction than the retrospective alternative. However, hospitals are heterogeneous with respect to their production technologies, making a random parameter frontier model the superior specification for Switzerland.

Appendix: Specifications for Bayesian estimation

This paper uses Bayesian inference to estimate Eqs. (1) and (3). Inference is made from a posterior distribution $p(\theta |X)$ of the unknown parameters (denoted by $\theta$) given the observed data (summarized as $X$). According to the Bayesian rule, this posterior distribution is given by

$$p(\theta |X)=\frac{ {\mathcal L}(X|\theta )p(\theta )}{p(X)}\propto p(\theta )\mathcal {L}(X|\theta ).$$

(9)

It is the product of the prior distribution $p(\theta )$ and the likelihood $\mathcal {L}(X|\theta )$, respectively.

For the estimates reported in Sect. 5, the posterior distribution pertaining to the SFM is specified as

$$\begin{aligned}&p(\alpha ,\beta , u,\gamma ,\sigma _v^{-2},\sigma _u^{-2};C,Y,W,Z) \propto p(\alpha ,\beta ,\gamma ,\sigma _v^{-2}, \sigma _u^{-2})\,\prod _{i=1}^N\prod _{t=1}^T p(u,\gamma ,\sigma _u^{-2}|Z)\nonumber \\&\qquad \times \prod _{i=1}^N\prod _{t=1}^T\frac{1}{\sqrt{2\pi \sigma _v^2}} \hbox{exp}\left[ -\frac{1}{2\sigma _v^2}\left( C_{it}-[C(Y_{it},W_{it};\alpha ,\beta ) +u_{it}]\right) ^2\right] , \end{aligned}$$

(10)

where $p(\alpha ,\beta ,\gamma ,\sigma _v^{-2}, \sigma _u^{-2})$ are probability distributions of the unknown parameters. The likelihood function in Eq. (10) is as in [16], i.e., normally distributed with $\sigma _v^2$ which denotes the variance of the random noise $v_{it}=C_{it}-[C(Y_{it},W_{it};\alpha ,\beta )+u_{it}]$. Equation (10) affords gain in flexibility over classical maximum likelihood estimation which requires a joint density function of the random noise and the inefficiency term to be specified. Here, only random noise enters the likelihood function. Inefficiency is estimated hierarchically as a latent variable along with the other parameters of the cost frontier.

Turning to the RPFM, the posterior is given by

$$\begin{aligned}&p(\alpha ,\bar{\alpha },\beta ,\bar{\beta }, u,\gamma ,\Sigma ,\sigma _v^{-2},\sigma _u^{-2};C,Y,W,Z) \propto p(\bar{\alpha },\bar{\beta },\gamma ,\Sigma ,\sigma _v^{-2}, \sigma _u^{-2})\,\prod _{i=1}^N \prod _{t=1}^Tp(u,\gamma ,\sigma _u^{-2}|Z)\nonumber \\&\qquad \times \prod _{j=1}^J (2\pi )^{-K/2}|\Sigma |^{-1/2}\hbox{exp}\left[ -\frac{1}{2}\left( {\alpha _{j} \atopwithdelims ()\beta _{j}}-{\bar{\alpha } \atopwithdelims ()\bar{\beta }}\right) '\Sigma ^{-1}\left( {\alpha _{j} \atopwithdelims ()\beta _{j}}-{\bar{\alpha } \atopwithdelims ()\bar{\beta }}\right) \right] \nonumber \\&\qquad \times \prod _{i=1}^N\prod _{t=1}^T\frac{1}{\sqrt{2\pi \sigma _v^2}} \hbox{exp}\left[ -\frac{1}{2\sigma _v^2}\left( C_{it}-[C(Y_{it},W_{it};\alpha _j,\beta _j) +u_{it}]\right) ^2\right] . \end{aligned}$$

(11)

Again, the likelihood function is specified as a normal distribution and inefficiency is estimated as a latent variable together with the other unknown parameters. The difference from Eq. (10) lies in the specification of the random intercept $\alpha _j$ and the slope parameters $\beta _j$, which are estimated at two levels. At the first level, overall determinants of hospital cost ($\bar{\alpha }$, $\bar{\beta }$) are taken into account, corresponding to the first factor following the proportionally sign of Eq. (11). The second-level estimates concern the effects ($\alpha _{j}$, $\beta _{j}$) associated with six hospital categories as defined in Eq. (4). They are derived from the multivariate normal distribution shown in Eq. (11).

In contrast to classical inference, Bayesian estimation requires information regarding the prior distributions of the unknown parameters, which are considered as random variables. These priors should comprise all information available before the sample data are used. In this paper, the values for the hyperparameters characterizing priors are chosen in a way to imply appropriate priors without imposing excessive restrictions. In particular, the priors for the SFM and RPFM are assumed to be independent,

$$\begin{aligned} p(\alpha ,\beta ,\gamma ,\sigma _v^{-2}, \sigma _u^{-2})&= p(\alpha ), p(\beta ), p(\gamma ), p(\sigma _v^{-2}), p(\sigma _u^{-2});\end{aligned}$$

(12)

$$\begin{aligned} p(\bar{\alpha },\bar{\beta },\gamma ,\Sigma , \sigma _v^{-2}, \sigma _u^{-2})&= p(\bar{\alpha }), p(\bar{\beta }), p(\gamma ),p(\Sigma ), p(\sigma _v^{-2}), p(\sigma _u^{-2}). \end{aligned}$$

(13)

Here, $p(\alpha )=f_N[0,\theta _{\alpha }]$, $p(\bar{\alpha })=f_N[0,\theta _{\bar{\alpha }}]$, $p(\beta )=f_N[0,\theta _{\beta }]$, $p(\bar{\beta })=f_N[0,\theta _{\bar{\beta }}]$ all have a normal distribution with mean zero and a diffuse prior for their corresponding variance $\theta$. The variance of the likelihood function has a gamma distribution $p(\sigma _v^{-2})=f_G[\mu ,\theta _{\sigma _v^{-2}}]$ with diffuse shape and scale parameters. Inefficiency is assumed to be truncated, with normal distribution $p(u,\gamma ,\sigma _u^2|Z)= f_N^+[\gamma Z,\sigma _u^2]$ and $\sigma _u^2= f_G[5, (5*log(\bar{r})^2)]$ and $p(\gamma )=f_N[0,\theta _{\gamma }]/\sqrt{f_G[5,(5*log(\bar{r})^2)]}$. This specification is in line with Griffin and Steel [16] and Koop et al. [19], permitting one to impose a priori information with regard to mean efficiency, $\overline{eff}=\hbox{exp}(-\overline{u})$. Again following Griffin and Steel [16], $\overline{eff}=0.875$ is assumed for prior efficiency. Finally, the variance of the random parameters is specified as a Wishart distribution $p(\Sigma )= f_W[S]$ in accordance with Eq. (11), with diffuse prior for the covariance matrix $S$.

Finally, note that estimates of the unknown parameters can be derived from the marginal posteriors of Eqs. (10) and (11). However, it is not always possible to compute posteriors analytically. Therefore, iterative Monte Carlo Markov Chain (MCMC) simulation is used, which involves iterative sampling from posterior parameter densities. Here, WinBUGS is used to derive these estimates (for an introduction, see [27]). The corresponding computational codes for the SFM and RPFM are shown in Table 4.

Table 4 Computation codes for the standard frontier model and the random parameter frontier model