Determinants of healthcare system’s efficiency in OECD countries

Abstract

Objective

Firstly, to compare healthcare systems’ efficiency (HSE) using two models: one incorporating mostly inputs that are considered to be within the discretionary control of the healthcare system (i.e., physicians’ density, inpatient bed density, and health expenditure), and another, including mostly inputs beyond healthcare systems’ control (i.e., GDP, fruit and vegetables consumption, and health expenditure). Secondly, analyze whether institutional arrangements, population behavior, and socioeconomic or environmental determinants are associated with HSE.

Design

Data envelopment analysis (DEA) was utilized to calculate OECD countries’ HSE. Life expectancy and infant survival rate were considered as outputs in both models. Healthcare systems’ rankings according to the super-efficiency and the cross-efficiency ranking methods were used to analyze determinants associated with efficiency.

Results

(1) Healthcare systems in nine countries with large and stable economies were defined as efficient in model I, but were found to be inefficient in model II; (2) Gatekeeping and the presence of multiple insurers were associated with a lower efficiency; and (3) The association between socioeconomic and environmental indicators was found to be ambiguous.

Conclusions

Countries striving to improve their HSE should aim to impact population behavior and welfare rather than only ensure adequate medical care. In addition, they may consider avoiding specific institutional arrangements, namely gatekeeping and the presence of multiple insurers. Finally, the ambiguous association found between socioeconomic and environmental indicators, and a country’s HSE necessitates caution when interpreting different ranking techniques in a cross-country efficiency evaluation and needs further exploration.

Appendix: the DEA methodology

Consider n DMUs, (countries in our case) where each DMU j (j = 1,…, n) uses m inputs $\vec{X}_{j} = \left( {X_{1j} ,X_{2j} , \ldots ,X_{mj} } \right)^{T} > 0$ to produce S outputs $\vec{Y}_{j} = \left( {Y_{1j} ,Y_{2j} , \ldots ,Y_{sj} } \right)^{T} > 0$. The CCR model is: For each unit k, we find the best weights $U_{r}^{k}$ $\left( {r = 1,2, \ldots ,S} \right)$ and $V_{i}^{k}$ $(i = 1,2, \ldots ,m)$ that maximize the ratio $h_{k} = \mathop {Max}\limits_{{U_{r}^{k} ,V_{i}^{k} }} \frac{{\sum\nolimits_{r = 1}^{S} {U_{r}^{k} Y_{rk} } }}{{\sum\nolimits_{i = 1}^{m} {V_{i}^{k} X_{ik} } }}\;\left( {k = 1, \, 2, \ldots , \, n} \right) \,$ (the ratio of total weighted output to the weighted input), where the weights $U_{r}^{k}$ and $V_{i}^{k}$ are nonnegative. The CCR model is formulated as follows:

$$ \begin{gathered} h_{k} = Max\sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rk}} } \hfill \\ s.t \hfill \\ \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ik}} = 1} {\text{ }} \hfill \\ \sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rj}} - \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ij}} } } \le 0\quad {\text{for}}\,{\text{ }}j = 1{\text{,}} \ldots {\text{,}}n \hfill \\ U_{r}^{k} \ge \varepsilon > 0\quad {\text{r}} = {\text{1,2,}} \ldots {\text{,}s} \hfill \\ V_{i}^{k} \ge \varepsilon > 0\quad {\text{i}} = {\text{1,2,}} \ldots {\text{,}m} \hfill \\\end{gathered} $$

(1)

$\varepsilon > 0$ is a non-Archimedean infinitesimal.

This model (CCR) assumes that the production function exhibits constant returns to scale (CRS). The BCC model developed by Banker, Charnes, and Cooper (BCC) in 1984 [25] adds a constant variable, $L_{k}$, to the weighted output $\left( {\sum\limits_{r = 1}^{S} {U_{r}^{k} Y_{rk} } } \right)$, in order to permit variable (decreasing, increasing) returns to scale (VRS). The output-oriented BCC model is formulated as follows:

$$ \begin{gathered} h_{k} = Max\sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rk}} } + L_{k} \hfill \\ s.t \hfill \\ \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ik}} = 1} {\text{ }} \hfill \\ \sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rj}} - \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ij}} } } + L_{k} \le 0\quad {\text{for}}\,{\text{ }}j = 1, \ldots ,n \hfill \\ U_{r}^{k} \ge \varepsilon > 0\quad {{r}} = {\text{1,2,}} \ldots {\text{,}s} \hfill \\ V_{i}^{k} \ge \varepsilon > 0\quad {{i}} = {\text{1,2,}} \ldots {\text{,}m} \hfill \\ \end{gathered} $$

(2)

The results of the CCR input-minimized or output-maximized formulations are the same, which is not the case in the BCC model [54]. Sueyoshi [28] presented eight types of efficiencies. In this paper, we considered the major two types of efficiencies: Technical efficiency (TE) and technical and scale efficiency (TSE). TSE is considered in the CCR model and TE in the BCC model.

The character $\varepsilon$ is a non-Archimedean quantity small number. The constraints $u_{r}^{A} \ge \varepsilon > 0$, $v_{i}^{A} \ge \varepsilon > 0$ are needed since the ordinary use of DEA frequently causes many multipliers to become zero, meaning that each input or output corresponding to a zero multiplier is not fully utilized for the DEA evaluation. This occurs because a DMU chooses only few inputs and outputs to optimize its score and disregards inputs and outputs, which would decrease its score. To deal with this problem, it is common to restrict the multipliers to a certain domain. However, there is no guarantee that this can always eliminate all of the zero multipliers because in many cases, this approach produces infeasible solutions. To overcome this disadvantage, Sueyoshi [28] and Sueyoshi and Goto [29] devised the slack-adjusted DEA (SA-DEA) for restricting the weights by only a lower bound, which is determined by the number of inputs and outputs. The suggestion was to divide each variable by its maximal value, namely

$$ {R_{i}^{ - } = \mathop {\max }\limits_{j} (X_{{ij}} )\quad {\text{for}}\quad i = {\text{1,2,}}...{\text{,}}m{\text{, }}R_{r}^{ + } = \mathop {\max }\limits_{j} (Y_{{irj}} )\quad {\text{for}}\quad r = {\text{1,2,}} \ldots {\text{,}}s} $$

(3)

whereupon, the bounds common to all inputs and outputs are:

$$ V_{i} \ge \left\{ {1/(m + s)R_{i}^{ - } } \right\}{\text{ }}\quad i = {\text{1,2,}} \ldots {\text{,}}m{\text{,}}\quad {\text{U}}_{r} \ge \left\{ {1/(m + s)R_{i}^{ + } } \right\}\quad r = {\text{1,2,}} \ldots {\text{,}}s $$

(4)

The super-efficiency ranking method

Andersen and Petersen (A&P) [30] view the DEA score for the inefficient DMUs as their rank scale. In order to rank scale the efficient DMUs, they suggest allowing the efficient DMUs to receive a score higher than one by dropping the constraint that bounds the score of the evaluated unit k, namely the primal problem of A&P of unit k will be formulated as follows:

$$\begin{gathered} h_{k} = Max\sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rk}} } + L_{k} \hfill \\ s.t \hfill \\ \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ik}} = 1} {\text{ }} \hfill \\ \sum\limits_{{r = 1}}^{s} {U_{r}^{k} Y_{{rj}} - \sum\limits_{{i = 1}}^{m} {V_{i}^{k} X_{{ij}} } } + L_{k} \le 0\quad {\text{ for }}j = 1,2, \ldots ,n,\quad {\text{ }}j \ne k \hfill \\ U_{r}^{k} \ge \varepsilon > 0\quad r = 1,2, \ldots ,s \hfill \\ V_{i}^{k} \ge \varepsilon > 0\quad i = 1,2, \ldots ,m \hfill \\ \end{gathered} $$

(5)

For more details, see Adler et al. [54].

The cross-efficiency

The cross-evaluation matrix was first developed by Sexton et al. [31]. This method calculates the efficiency score of each DMU n times, using the optimal weights evaluated at each run. The results of all the DEA cross-efficiency are summarized in a matrix as given:

$$ h_{k,j} = \frac{{\sum\nolimits_{r = 1}^{s} {U_{r}^{k} Y_{rj} } }}{{\sum\nolimits_{i = 1}^{m} {V_{i}^{k} X_{ij} } }}\quad j = 1,2, \ldots ,n\quad k = 1,2, \ldots ,n $$

Thus, $h_{{_{kj} }}$represents the score given to DMU j by the optimal weights of DMU k. The elements on the diagonal $h_{{_{kk} }}$represent the standard DEA scores $h_{k}$. The cross-efficiency ranking method utilizes the matrix $h_{{_{kj} }}$ to rank the units on one scale. If we define $\bar{h}_{k} = \frac{{\sum\nolimits_{j = 1}^{n} {h_{kj} } }}{n}$ as the average cross-efficiency score for DMU k, one can rank all the DMUs on one scale according to $\bar{h}_{k}$. The $\bar{h}_{k}$ score utilizes the weights of all other DMUs, and therefore, all inputs and outputs for each DMU are evaluated with the same set of weights. Thus, one can rank the DMUs on the same scale by $\bar{h}_{k}$.