Efficiency of New Zealand’s District Health Boards at Providing Hospital Services: A stochastic frontier analysis

Abstract

The majority of secondary and tertiary healthcare services in New Zealand are provided through public hospitals managed by 20 local District Health Boards. Due to data issues and ill-judged generic public perceptions, efficiency studies are insufficient in spite of the extensive empirical literatures available. This inevitably leads to criticisms about the perverse incentives which might be created by the National Health Targets designed to improve the performance of public health services. Utilizing a multifaceted administrative hospital dataset, this is the first case study to measure both the technical and cost efficiency of New Zealand public hospitals during the period of 2011–2017. More specifically, it deals with the question of how hospital efficiency varies with respect to activities accounted for by the National Health Targets. The empirical results show no evidence that these targets are achieved at the expenses of lowering the overall efficiency of hospital operations. The national technical efficiency is averaged at 86 percent over the period and cost efficiency is 85 percent. The results are derived by stochastic input distance function and cost frontier in order to accommodate multiple outputs and limited number of census observations. Efficiency ranking is sensitive to specifications of the inefficiency error term, but reasonably robust to the choice of functional form and different proxies for capital input.

Appendices

Appendix A

See Table 5

See Table 5 New Zealand DHB

Appendix B

See Table 6

See Table 6 Health targets for 2017

Appendix C

See Table 7

See Table 7 Translog input distance function and cost frontier estimates

Appendix D: The DEA Model

With price information and under the behavioural objective of cost minimization, both technical and cost efficiencies can be measured using the standard DEA model as outlined in Färe et al. (1993) and the software DEAP 2.1 developed by Coelli 1996).

First the input-oriented DEA model is run to obtain technical efficiencies (i.e. TE_DEA in Table 4), assuming there is data on K inputs and M outputs on each of N firms or decision making units (DMUs). For the i-th DMU these are represented by the vectors x_i and y_i, respectively. The K × N input matrix, X, and the M × N output matrix, Y, represent the data of all N DMU’s. The purpose of DEA is to construct a non-parametric envelopment frontier over the data points such that all observed points lie on or below the production frontier. This is accomplished by solving the corresponding variable returns to scale (VRS) linear programming problem:

$$\begin{array}{l}{\mathrm{min}}_{\theta ,{\boldsymbol{\lambda }}}\,\theta ,\\ {\mathrm{st}}\ - {\boldsymbol{y}}_{\boldsymbol{i}} + {\boldsymbol{Y\lambda }} \ge 0,\\ \theta {\boldsymbol{x}}_{\boldsymbol{i}} - {\boldsymbol{X\lambda }} \ge 0,\\ {\mathbf{N}}1\prime {\boldsymbol{\lambda }} = 1,\\ {\boldsymbol{\lambda }} \ge 0,\end{array}$$

where θ is a scalar and λ is a N × 1 vector of constants for all NDMUs. N1 is an N × 1 vector of ones. The value of θ obtained will be the efficiency score for the i-th DMU. It will satisfy θ ≤ 1, with a value of 1 indicating a point on the frontier and hence a technically efficient DMU. Note that the linear programming problem must be solved N times, once for each DMU in the sample. A value of θ is then obtained for each DMU.

Next the following cost minimization DEA model is run to obtain cost efficiencies (i.e. CE_DEA in Table 4):

$$\begin{array}{ll}&{\mathrm{min}}_{{\boldsymbol{\lambda }},{\boldsymbol{x}}_{\boldsymbol{i}}^ \ast }\,{\boldsymbol{w}}_{\boldsymbol{i}}^\prime {\boldsymbol{x}}_{\boldsymbol{i}}^ \ast ,\\ &{\mathrm{st}} - {\boldsymbol{y}}_{\boldsymbol{i}} + {\boldsymbol{Y\lambda }} \ge 0,\\ &{\boldsymbol{x}}_{\boldsymbol{i}}^ \ast - {\boldsymbol{X\lambda }} \ge 0,\\ &{\mathbf{N}}1\prime {\boldsymbol{\lambda }} = 1,\\ &{\boldsymbol{\lambda }} \ge 0,\end{array}$$

where w_i is a vector of input prices for the i-th DMU and ${\boldsymbol{x}}_{\boldsymbol{i}}^ \ast$ (which is calculated by the Linear Programming) is the cost minimizing vector of input quantities for the i-th DMU, given the input prices w_i and the output levels y_i. The total cost efficiency (CE) of the i-th DMU would be calculated as:

$${\mathrm{CE}} = \frac{{{\boldsymbol{w}}_{\boldsymbol{i}}^\prime {\boldsymbol{x}}_{\boldsymbol{i}}^ \ast }}{{{\boldsymbol{w}}_{\boldsymbol{i}}^\prime {\boldsymbol{x}}_{\boldsymbol{i}}}}$$