Falling Short on Long-Term Care Efficiency Change? A Non-Parametric Approach

Abstract

The European Commission’s 2015 aging report forecasts a substantial increase in public spending on Long-Term Care (LTC) for OECD countries by 2060, posing significant fiscal challenges. This study aims to assess the efficiency and productivity of the LTC sector from 2010 to 2019 and explore whether efficiency gains can alleviate these fiscal pressures. Using a non-parametric Data Envelopment Analysis (DEA) model, combined with Tobit regression, we estimate the efficiency of OECD countries and examine the role of decentralization in shaping performance outcomes. The findings reveal that, on average, countries operate at 94% efficiency, with modest productivity growth. However, technical inefficiencies persist, especially in unitary countries, while federal countries, though initially less efficient, show greater improvements over time. Despite these gains, the current efficiency levels are insufficient to counterbalance the projected increase in LTC demand. Policymakers should prioritize reforms that enhance efficiency through decentralization, promoting accountability and competition as mechanisms to sustain the LTC system in the face of demographic shifts.

1. Introduction

The Organization for Economic Cooperation and Development (OECD) defines Long-Term Care (LTC) as a range of services provided to people who have functional limitations or chronic conditions that make it difficult for them to perform basic activities of daily living (ADLs) (such as bathing, dressing, eating, and toileting) for an extended period of time (Colombo et al. 2011). These services may be provided in various settings, such as the individual’s home, a residential care facility, or a hospital, and may include assistance with personal care, medical and nursing care, rehabilitation, social support, and assistance with instrumental activities of daily living (IADLs). IADLs also comprise managing finances, cooking, and shopping. LTC is primarily focused on maintaining or improving the quality of life and functional status of people with chronic conditions, rather than providing acute medical intervention.

Siciliani (2013), along with Cremer (2014) and the recent findings from Verbakel et al. (2023), collectively emphasize that LTC presents challenges that primarily, but not exclusively, affect older individuals.

According to Österle (2024), beyond the OECD definition, other definitions also exist, differing in content and scope. Nonetheless, they each aim to address three key dimensions of the issue: the needs associated with chronic diseases, the scope of services provided, and the duration over which these services are offered.

The provision of Long-Term Care (LTC) services is a critical issue in countries belonging to the OECD due to the aging population and the increasing demand for medical and social care. According to the OECD, the share of the population aged 65 and over is projected to increase from 17% in 2020 to 27% in 2050, and 71% of people 65 years and older will be in middle-income countries (UN DESA et al. 2020). As a result, the demand for LTC services is expected to rise significantly in the coming decades. However, the availability and quality of these services vary widely across countries, many of which are facing significant challenges in ensuring adequate and affordable care for their aging populations (Fernández and Gori 2016).

At their inception between the 1970s and 1990s, LTC services were a “residual” policy provided at the local and state/regional levels to meet the demand for these services, supporting those provided by the health system. Since the late 1990s, there has been an overall tendency to make LTC universal through the social security system or tax-based frameworks (Ranci and Pavolini 2013; Theobald and Ozanne 2015; Gori et al. 2016). The shift in the jurisdiction of LTC provision in various countries has been driven by several factors. In Sweden, rising costs and market-oriented policies have led to a more centralized approach, reducing generosity and increasing outsourcing. Germany has integrated LTC policies within a multi-level governance framework yet faces challenges in local integration due to market-oriented care provision. Overall, these changes aim to manage costs, integrate market dynamics, and adapt to existing governance and funding structures, although significant integration and funding challenges remain.

In 2021, on average, LTC expenditure accounted for 1.5% of gross domestic product (GDP) across the OECD (OECD 2021). In the future, it is expected that average healthcare spending on LTC will increase to 3% of GDP in 2060 (European Commission et al. 2015). De la Maisonneuve and Martins (2014) estimate a significant increase, anticipating that some OECD countries will experience a two- to three-fold rise in their ratios of public LTC spending relative to GDP by 2060.

Cremer (2014) assesses that there are three primary institutions responsible for financing and delivering LTC services: the family, the market, and the state. In many OECD countries, provision costs are financed through a public funding mechanism because insurance markets are either not allowed or have not been developed. In addition, the supply of informal care is potentially shrinking due to several factors, including changes in family structure, increased female labor force participation, and migration (European Commission 2021). As families become smaller and more dispersed, there may be fewer potential caregivers available to provide informal care to older adults. In addition, as more women enter the workforce, they may have less time available to provide care. Finally, migration can also impact the availability of informal caregivers, as individuals may move away from family members in search of better job opportunities or improved quality of life (OECD 2019). Muir (2017) assess that unless family and friends can provide informal care, many people will be unable to afford LTC in their own home, leaving them with unmet needs or at risk of early institutionalization. This has contributed to a rising burden on public finances due to greater spending on LTC services. For example, from 2004 to 2019, average government expenditure on LTC doubled as a share of gross domestic product (GDP), growing nearly four times faster than health spending in a period where all government expenditure-to-GDP ratios slightly contracted (De Biase and Dougherty 2023). Across OECD countries, USD four out of five spent on LTC come from public sources (OECD 2021). Consequently, the future fiscal sustainability of LTC provision is a pressing concern for many developed nations. To address this challenge, policymakers must gain a comprehensive understanding of the dynamics of public LTC spending to formulate policies that ensure the enduring financial viability of LTC services. Such trends will entail raising taxes, cutting expenses, reallocating public budgets, reducing the quality of provisions or (de)centralizing service provisions. In the past, combinations of these solutions have involved marketization and privatization, re-familiarization, or public support becoming more targeted towards those with intense care needs (Da Roit 2021; Rostgaard et al. 2022).

Alternatively, one way to mitigate the pressure on public budgets is through the application of best practices. In this article, we perform efficiency analysis and calculate productivity changes in the LTC sector in OECD countries spanning the 2010–2019 period in order to detect good practices and potential efficiency gains that could later be translated into lower costs or a higher level of service provision given the resources. This is in line with New Public Management (NPM) (Hood 1991; Peters and Pierre 1998; Dan and Pollitt 2015). NPM can be considered a mechanism for cutting costs and achieving more for less as a result of better-quality management and different structural design (Osborne 1993; Gore 1993). In addition, we explore differences in efficiency and productivity for countries with federal or unitary organization to evaluate whether the political–administrative organization of the countries has any influence.

This paper contributes to the empirical literature by combining LTC and productivity analysis. On the one hand, we take previous studies on LTC services to characterize the service provision; namely, the inputs and the outputs that affect the production function. On the other hand, we take the best models that allow for the estimation of changes in productivity from the literature related to technical efficiency.

In the following sections, we will provide a concise review of the relevant literature (Section 2). Afterwards, we describe the methodology (Section 3) and data and variables (Section 4). We then present the Data Envelopment Analysis results for the technical efficiency and the productivity change (Section 5). Finally, we present our conclusions in Section 6.

2. Background

The first comprehensive review of the economics of LTC was conducted by Norton (2000), who differentiated LTC from acute medical care across four dimensions: its focus on chronic illnesses, the prevalence of for-profit nursing home facilities, the significant role of unpaid caregivers, and the limited adoption of private LTC insurance. De la Maisonneuve and Martins (2014) added that, unlike healthcare services aiming to improve health conditions, LTC primarily aims to make chronic conditions more bearable.

Following De la Maisonneuve and Martins (2014), the expenditure on LTC depends on two kinds of determinants: demographics and non-demographics. The first one is related to the number of old dependent people in the population and other dependents. In turn, this factor depends on life expectancy and health expenditure. It is expected that scientific progress in areas such as medicine and health in OECD countries will cause life expectancy to increase significantly, as well as the proportion of people over the age of 80 and the life expectancy of people with disabilities. The European Union (EU) anticipates a noteworthy increase in the old-age dependency ratio, rising from 29.6% in 2016 to 51.2% by 2070. This signifies a shift from 3.3 working-age individuals per person aged 65 and above in 2016 to only 2 by 2070. The surge is particularly pronounced in the very-old-age dependency ratio (individuals aged 80 and above), which is projected to increase from 8.3% in 2016 to 22.3% in 2070. Consequently, a substantial growth in demand for related services is expected in the future. In this sense, it is not only the number of dependents that matters, but it is also crucial to determine if increased life expectancy correlates with a higher prevalence of disability. Many studies conclude that both age and time to death play a role in predicting LTC expenditure (Murtaugh et al. 1990; Rudnytskyi and Wagner 2019; Siciliani 2013).

Within the non-demographic determinants are the average income of the population, the low productivity of the sector, and changes in the informal provision of LTC (OECD 2020). As average income increases, this has a direct impact on the standard of living and the associated LTC demand. According to Serrano-Alarcón et al. (2021), there is a growing demand for medical care caused by an increase in comorbidities and life expectancy among the population. In turn, there is an indirect effect of increased average income through productivity. That is, as income increases, the dynamic sectors of the economy become more productive, and this increases the wages of those workers. In order to retain workers in the LTC sector, their wages should rise to match those of other sectors, even though productivity in that sector has not changed (Baumol effect). The supply side of LTC services is characterized by its labor-intensive nature, with services being delivered by both formal and informal caregivers (Ozbugday et al. 2020). Formal caregivers encompass paid professionals, including nurses and individuals offering essential personal care, such as bathing, dressing, or grooming, to those in need due to factors such as aging, illness, injury, or other physical and mental conditions. These services can be provided in private homes or institutional settings, excluding hospitals. This category also includes family members, neighbors, or friends who are employed under a formal contractual agreement and/or registered with social security systems as caregivers. Conversely, informal caregivers consist of family members, friends, or individuals offering assistance without formal employment or compensation. Studies on the public provision of home care have revealed that formal care may partially substitute informal care, although at a notably lower rate (Christianson 1988; Moscovice et al. 1988; Pezzin et al. 1996; Bonsang 2009; Bolin et al. 2008; Perdrix and Roquebert 2020). This implies that formal spending on LTC is a good indicator of the total expenditure (formal and informal) on LTC in each country. In addition, on the one hand, informal care is decreasing due to changes in family structure, increased female labor force participation, and migration.

As mentioned in the Introduction, there are three primary institutions responsible for financing and delivering LTC services: the family, the market, and the state. However, USD four out of five spent on LTC comes from public sources, implying that most of the expenditure is through the public sector. One of the reasons for the almost exclusive participation of the public sector in providing LTC services is market failures. Norton (2000) argues that market failures, such as information asymmetry and adverse selection, can lead to inefficiencies and inequities in LTC provision, necessitating government intervention to address them. Siciliani (2013), Cremer (2014), and Klimaviciute and Pestieau (2018) provide a more recent review of the economics of LTC, focusing on the challenges faced by policymakers discussing various policy options: public provision, private provision with public financing (Private–Public Partnership), and public insurance. A related topic is that the type of assistance has different implications. Verbakel et al. (2023) assess that different types of LTC policies on providing in-kind or in-cash LTC services or facilitating care provision for families have different socio-economic impacts on inequalities in healthcare. Overall, the literature on LTC economics highlights the need for policy interventions to address market failures and promote efficiency, quality, and equity in the provision. The specific policy options and trade-offs will depend on the context and the specific objectives of policymakers.

Organizing the information differently, Figure 1 characterizes the sector by indicating that, from a supply perspective, LTC services are financed through three main sources: families, the market, and the state. These sources contribute varying levels of resources to support a diverse range of services that require personnel hours (both formal and informal), infrastructure, and other materials or resources. Demand for LTC services, on the other hand, is characterized by two types of recipients: those residing at home and those in institutional settings. This demand is influenced by life expectancy and the quality of services required, as well as other drivers, such as the dependency ratio, which reflects the sector’s relative weight within society, and cultural patterns.

Efficiency analyses of care services at the national level have been a popular research topic since the 1990s, while Public Management Reform was a high issue on the agenda (Hood 1991; Barzelay 1992; Osborne 1993; Gore 1993; Peters and Pierre 1998). As far as NPM is seen as a solution for deprived and ineffective public services, it has created an impetus for cultural change towards results-based public management (Dan and Pollitt 2015), including efficiency analysis as a central core (Blomqvist 2004; Pollitt and Bouckaert 2011).

Nevertheless, most studies have been conducted in healthcare (Moberg et al. 2016), while LTC studies are still ongoing (Ogden and Adams 2009; Ansah et al. 2017; Mosca et al. 2017; Ozbugday et al. 2020; Kordic and Visic 2023; Zhang et al. 2023; Sevim et al. 2024; Wende et al. 2024).

This situation could be partially attributed to the accessibility of OECD health data, which have served as a catalyst for empirical research examining the efficiency of healthcare systems. Empirical studies on LTC demand focus on nursing home care. In OECD countries, more than half of health and social LTC spending in 2019 was allocated to nursing homes (OECD 2021). The predominant factor influencing LTC demand is an individual’s health status, encompassing both physical and mental health.

Varabyova and Müller (2016) conducted a meta-analysis encompassing studies across OECD countries, critically assessing the methods used, the results obtained, and the validity of the efficiency estimates. In contrast, Mbau et al. (2023) reviewed efficiency studies at national and subnational health levels, finding that most studies (60%) focused on national levels, 94% conducted quantitative analyses, and Data Envelopment Analysis (DEA) was the predominant non-parametric method used, followed by stochastic frontier analysis as the parametric method.

Many studies have used non-parametric methods like DEA to measure healthcare system efficiency in countries including Spain (Aparicio et al. 2014), India (Bhat 2005), and China (Cheng and Zervopoulos 2014). Others employed parametric methods, often econometric models, to estimate factors affecting healthcare efficiency (Gravelle et al. 2003; Wranik 2012).

In addition to measuring technical efficiency, some studies have also investigated the factors that affect healthcare efficiency. For example, Wranik (2012) identified best practices in primary care for chronic disease patients, while de Cos and Moral-Benito (2014) investigated the impact of Information and Communication Technologies (ICTs) and Research and Development (R&D) on healthcare efficiency in OECD countries.

To the best of our knowledge, Ozbugday et al. (2020), Kordic and Visic (2023), Wende et al. (2024), and Sevim et al. (2024) are the only studies that analyze cross-country efficiency changes within the LTC sector. Our article shares some similarities with Ozbugday et al. (2020), Kordic and Visic (2023), and Sevim et al. (2024): all use data from the OECD and perform a radial DEA model to calculate efficiency scores and productivity changes. However, our article differs in several aspects from theirs, thus allowing for new insights into the problem. First of all, regarding the setting, our panel comprises sixteen countries in ten years (2010–2019) instead of the seventeen countries in six years (2009–2014) in Ozbugday et al. (2020), twelve countries in six years (2014–2019) in Kordic and Visic (2023), and fifteen countries in 2019 in the case of Sevim et al. (2024). This increase in the data sample (especially in the longitudinal dimension) allows for a better understanding of the productivity change.

Secondly, from the technology point of view, Ozbugday et al. (2020) use two inputs (expenditure and total beds in residential care facilities) and one output (recipients in institutions). Indeed, despite the labor-intensive nature of the LTC sector, the study faced a significant limitation in that it could not incorporate the variable of the total number of workers as an input. Ozbugday et al. (2020) recognize this limitation attributed to the deficiency in available data on the workforce at the time the article was written. Kordic and Visic (2023) employ beds in residential LTC facilities and formal LTC workers in institutional settings as inputs, with institutional LTC recipients as the output. This approach, however, does not account for the provision of home-based services

We consider three inputs (we add nurses and personal caregivers) and two outputs (we separate recipients into recipients in institutions and recipients at home). By including the total number of workers, we include the main input in the production function; by distinguishing recipients at home from those in institutions, we can address a possible omitted variable problem. Regarding this matter, there is a notable difference in the number of LTC recipients receiving care at home compared to those in institutional settings. Specifically, the number of individuals receiving LTC at home is three times greater than those in institutions. Notably, in countries such as Norway, Denmark, and Belgium, which allocate significant resources to LTC, ambulatory home care providers play a crucial role, accounting for approximately 40% or more of the total LTC spending. In contrast, in countries like Estonia and Lithuania, these providers play a comparatively minor role in LTC provision.

Thirdly, Ozbugday et al. (2020), Kordic and Visic (2023), and Sevim et al. (2024) use an input-oriented DEA model, while we use an output-oriented one. We have chosen this orientation because we are interested in knowing how many more recipients could be served with a given amount of resources. To illustrate this point, it is worth noting that the OECD (2021) evaluation reveals that nearly two-thirds of LTC spending in Korea is directed toward hospital-based care. Hospitals also play a substantial role as LTC providers in Japan, accounting for 24% of total spending, as well as in Canada, Spain, Estonia, and Latvia, where they represent 12–16% of LTC expenditure. While hospital-like facilities have historically been involved in LTC delivery in Japan and Korea, a significant share of LTC funding being allocated to hospitals could signal potential deficiencies in alternative, lower-level care facilities for individuals with inpatient LTC requirements in certain countries. In the case of the United States, Norton (2000) underscores how certificate-of-need regulations have restricted the availability of nursing home beds in various U.S. states, resulting in waiting lists that contribute to the inefficient production of nursing home care and that have adverse implications for the quality of care provided. In Spain, Peña-Longobardo et al. (2016) assess that in 2006 a new System for Promotion of Personal Autonomy and Assistance for Persons in situation of Dependency (SAAD) was created, where, according to the severity of the case, individuals were progressively allowed to access the benefits. However, the 2008 crisis delayed the applications of the beneficiaries. As a result of that, there was a “dependency limbo”: persons officially assessed as entitled to benefits who had not actually received any (in-kind or monetary) provisions. In the case of Portugal, Lopes et al. (2018) assess that despite all inhabitants being eligible for LTC, the existence of regional asymmetries in care coverage still poses an important barrier to access to LTC and that 93% of the Portuguese population had poor access to institutionalized care in 2014, given the lack of beds available. Finally, Albuquerque (2022) characterizes people over 50 years of age in southern Europe, distinguishing whether or not they need LTC, and if they do, whether those needs are met formally, informally, or not met at all. She finds that as people become older the probability of unmet needs decreases and for older ages provision is more likely to be formal.

Additionally, we explore the technical efficiency of the inefficient countries and provide an analysis of efficient benchmark countries to help them improve their efficiency.

Given that, as Theobald and Ozanne (2015) state, it was initially a “residual” policy provided at the local and state/regional levels, we separated the sample of countries into unitary and federal to evaluate whether efficiency or productivity changes are associated with political–administrative organization. Unitary countries might benefit from economies of scale and the ability to adjust to idiosyncratic shocks. Conversely, federal countries’ decentralized provision at subnational levels could serve as a mechanism of yardstick competition (Shleifer 1985), making it easier to evaluate decision quality. Additionally, decentralized provision could adjust principal–agent problems between constituents and incumbents (Besley 2007).

3. Methodology

There are two methods for estimating efficiency: the parametric and the non-parametric approach (Coelli et al. 2005). In the first case, the researcher proposes a functional form to relate inputs and outputs in an efficient way. The dominant parametric method is stochastic frontier analysis (SFA), introduced by Aigner et al. (1977), and Meeusen and van Den Broeck (1977), in which a production function is estimated and then the residual of the estimated regression is separated into an efficiency component and an error term.

In the second case, the non-parametric approach uses linear programming to establish the production frontier, allowing productivity and efficiency to be calculated. DEA is the predominant non-parametric method. The first model was developed by Charnes et al. (1978) based on the work of Farrell (1957).

Depending on the data used, both methods can be used to calculate technical efficiency. Since our sample is relatively small and the SFA method requires many observations to be able to estimate a flexible production function, we consider the DEA method to be more appropriate. Specifically, our dataset comprises a panel of 16 countries over a 10-year period. Estimating a translog distance function would require at least 21 parameters, excluding those needed to control for unobserved heterogeneity. In addition, DEA offers several advantages for our specific context. It does not rely on assumptions concerning the statistical properties of the variables, which can be especially beneficial with limited data. Furthermore, DEA avoids multicollinearity problems and can accommodate multiple inputs and outputs without the need to specify a particular functional form for the production frontier, making it a pragmatic choice for our analysis (Avkiran and Zhu 2016 or Coelli et al. 2005).

Once the DEA approach is defined, the method presents two fundamental alternatives: input-oriented and output-oriented models. In the former, the primary aim is to minimize input usage while keeping output constant. Conversely, output-oriented models seek to maximize output with the given level of input.

Let us define an input vector as $X=\left(x_1,\dots ,x_m\right)$, and an output vector as $Y=\left(y_1,\dots ,y_p\right)$. We have n decision-making units (DMUs) that use m inputs to produce p outputs, denoted as $\left(X_j,Y_j\right),$ $j=1,\dots ,n$.

The output-oriented DEA model under the assumption of Constant Return to Scale (CRS) introduced by Charnes et al. (1978), referred to as the CCR model, is expressed as follows:

$$\begin{array}{llll}\hfill & \underset{{\varphi }_{c0},{\lambda }_{j0}}{Max}\hfill & {\varphi }_{c0}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & {\displaystyle \sum _{j=1}^n{\lambda }_{j0}{x}_{ij}}\le x_{i0},\hfill & i=1,\dots ,m\hfill \\ \hfill & \hfill & {\displaystyle \sum _{j=1}^n{\lambda }_{j0}{y}_{rj}}\ge {\varphi }_{c0}{y}_{r0},\hfill & r=1,\dots ,p\hfill \\ \hfill & \hfill & {\lambda }_{j0}\ge 0,\hfill & j=1,\dots ,n\hfill \end{array}$$

(1)

$\left(X_o,Y_o\right)$ is the input–output vector of the DMU₀. ${\varphi }_{co}$ is the technical efficiency of the DMU₀. The inverse of the optimal value (${\varphi }_{co}$) of model (1) coincides with $D_c\left(X_o,Y_o\right)={\displaystyle \frac{1}{{\varphi }_{co}}}$, the relative technical efficiency of the DMU₀ measured through the Shephard distance function output orientation (Shephard 1953). The range for $D_c\left(X_o,Y_o\right)$ is between 0 and 1, and a greater $D_c\left(X_o,Y_o\right)$ means greater technical efficiency. On the limit, the DMU₀ is efficient if $D_c\left(X_o,Y_o\right)=1$ because it means there is no room for output $Y_o$ expansion with the $X_o$ vector of inputs. ${\lambda }_{j0}$ are the intensity variables.

If we restrict all lambdas to sum up to $1$, we obtain the output-oriented DEA model under the assumption of Variable Return to Scale (VRS) introduced by Banker et al. (1984), referred to as the BCC model. ${\varphi }_{v0}$ is the technical efficiency of the DMU₀, obtained from the BCC model. In our case, given that the countries in our dataset vary significantly in size, we consider the BCC model to be more appropriate, as it allows for the possibility that the technology available to smaller countries may differ from that available to larger countries.

We assess the productivity change in our sample by implementing the output-oriented Malmquist Index proposed by Caves et al. (1982) and Färe and Grosskopf (1992). The Malmquist Index is a non-parametric productivity index derived from Data Envelopment Analysis (DEA). It allows for the estimation of changes in total factor productivity for all the DMUs over the years.

The output-oriented $M_c$ for productivity change for DMUs (countries in our case) from period $t$ to period $t+1$ under Constant Returns to Scale can be defined as follows:

$$M_c\left(X_o^{t+1},Y_o^{t+1},X_o^t,Y_o^t\right)={\left[{\displaystyle \frac{D_c^t(X_o^{t+1},Y_o^{t+1})}{D_c^t(X_o^t,Y_o^t)}}\times {\displaystyle \frac{D_c^{t+1}(X_o^{t+1},Y_o^{t+1})}{D_c^{t+1}(X_o^t,Y_o^t)}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(2)

where $D_c^s\left(X_0^h,Y_0^h\right)$ is the Shephard output distance function calculated from the period $h$ observation ($X_0^h,Y_0^h$), $h=t$,$t+1$, to the frontier of the technology at time $s$, $s=t$,$t+1$ under CRS, and $D_v^s\left(X_0^h,Y_0^h\right)$ under VRS.

Improvements in productivity yield Malmquist Productivity Indexes greater than unity. Deterioration in performance over time is associated with a Malmquist Productivity Index less than unity.

On the other hand, the scale efficiency of $\left(X_o^h,Y_o^h\right)$ is measured through the ratio of

$${SE}^s\left(X_o^h,Y_o^h\right)={\displaystyle \frac{D_c^s(X_o^h,Y_o^h)}{D_v^s(X_o^h,Y_o^h)}}$$

(3)

It holds that $S{E}^s\left(X_o^h,Y_o^h\right)\le 1$ and, if $S{E}^s\left(X_o^h,Y_o^h\right)=1$, it means that the DMU₀ producing $Y_o^h$ with the input vector $X_o^h$ is positioned in the most productive scale size region of the technology. Hence, $S{E}^s\left(X_o^h,Y_o^h\right)$ is an efficiency indicator showing how far $(X_o^h,Y_o^h)$ deviates from the point of technically optimal scale of operation.

Ray and Delsi (1997) decompose the $M_c\left(X_o^{t+1},Y_o^{t+1},X_o^t,Y_o^t\right)$ as follows:

$$\begin{array}{ll}{M}_c\left(X_o^{t+1},Y_o^{t+1},X_o^t,Y_o^t\right)=& \\ & \underset{\begin{array}{c}Technical Efficiency \\ Change \left(TEC\right)\end{array}}{\underbrace{{\displaystyle \frac{D_v^{t+1}(X_o^{t+1},Y_o^{t+1})}{D_v^t(X_o^t,Y_o^t)}}}}\times \underset{\begin{array}{c}Technologial Efficiency\\ Change \left(TC\right)\end{array}}{\underbrace{{\left[{\displaystyle \frac{D_v^t(X_o^{t+1},Y_o^{t+1})}{D_v^{t+1}(X_o^{t+1},Y_o^{t+1})}}{\displaystyle \frac{D_v^t(X_o^t,Y_o^t)}{D_v^{t+1}(X_o^t,Y_o^t)}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}\\ & \times \underset{\begin{array}{c}Scale Efficency \\ Change \left(SEC\right)\end{array}}{\underbrace{{\left[{\displaystyle \frac{{SE}^t(X_o^{t+1},Y_o^{t+1})}{{SE}^t(X_o^t,Y_o^t)}}{\displaystyle \frac{{SE}^{t+1}(X_o^{t+1},Y_o^{t+1})}{{SE}^{t+1}(X_o^t,Y_o^t)}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}\end{array}$$

(4)

The equation above shows that the productivity change $M_c$ is the product of three sources of improvement: the first source is the technical efficiency change ($TEC$), which captures whether actual production is moving closer (catching up) or farther from the efficient production on the frontier. The second one is the technological efficiency change ($TC$), which captures the shift in technology between the two periods evaluated and is intended to capture innovation or the frontier shift. The third one is the scale efficiency change (SEC).

4. Data and Variables

To perform the analysis, we use data from OECD Health Statistics (OECD 2023). The OECD Health Database offers the most comprehensive source of comparable statistics on health and health systems across OECD countries. OECD has been compiling data on LTC since 1995. As countries have been gradually adding their information, there may be gaps. On the other hand, some countries have not submitted information or have just provided it for a few years either because it is not available or is not consistent with the rest of the countries. For example, the range and comparability of LTC spending estimates has continuously improved with the implementation of “A System of Health Accounts 2011” and “Accounting and mapping of Long-Term Care Expenditure under SHA 2011” which provide a framework for the measurement of health and LTC spending and a separation of health and social spending (OECD/Eurostat/WHO 2017).

For this reason, our sample is a balanced panel data for 16 countries (Austria, Canada, Denmark, Estonia, Germany, Hungary, Israel, Korea, Luxembourg, the Netherlands, Norway, Slovakia, Spain, Sweden, Switzerland, United States) for the years 2010 to 2019. Each of these countries is considered a DMU. We acknowledge that in some countries, service provision is managed by subnational government levels, meaning the national government is not strictly the decision-making unit. However, to make the results comparable, the national level has been maintained as the unit of analysis. Moreover, the OECD does not provide data at the subnational level, and if such data existed, unitary countries would still have a single observation while federal countries would have multiple observations, thus biasing the representativeness. Some countries where substantial progress has been made in the provision of LTC services, such as Australia and Portugal, were not included in the analysis due to the lack of available data on all variables for the period considered.

Table 1. Descriptive statistics.

Variable	Variable Definition	N	Mean	St. Dev	Min	Max
(Y1)	Recipients in institutions	160	223,875	354,962	3881	1,400,810
(Y2)	Recipients at home	160	681,272	1,161,566	7966	5,395,496
(X1)	Formal workers	160	335,918	642,419	4912	2,861,973
(X2)	Beds in LTC facilities	160	259,324	421,230	3881	1,663,445
(X3)	Expenditure per capita (PPP)	160	721.00	524.00	6.00	1991.00

provides summary statistics for the variables included. We considered two outputs, “Recipients in institutions” (other than hospitals) ($Y_1$) and “Recipients at home” ($Y_2$). In both scenarios, our analysis encompasses individuals who receive LTC services from paid providers, comprising both professional caregivers and non-professionals who receive compensation through various means. This includes individuals receiving cash payments under social programs, as well as beneficiaries of cash benefits like consumer-choice programs, care allowances, or other social benefits. These financial aids are typically granted with the primary objective of providing support to individuals with LTC needs, based on a comprehensive assessment of their specific care requirements. The definition excludes people receiving informal LTC services from relatives or informal workers.

We considered three inputs. The first one is “Formal workers” ($X_1$). LTC workers encompass two primary categories of professionals: (i) Nurses: these are healthcare professionals who provide LTC services either in private homes or LTC institutions (excluding hospitals); (ii) Personal Care Workers (Caregivers): These are individuals who offer formal LTC services, both in private homes and non-hospital institutions, but they do not hold nursing qualifications or certifications.

The second input is “Beds in LTC facilities” ($X_2$); these facilities comprise establishments primarily engaged in providing residential LTC that combines nursing, supervisory, and other types of care as required by the residents.

Finally, the third input, “Expenditure per capita” ($X_3$), is the expenditure from all financing schemes in per capita current prices converted into PPP units to account for differences in price levels across countries.

According to the OECD (2021), LTC spending comprises both health and social services to LTC-dependent people who need care on an ongoing basis. Based on the System of Health Accounts (OECD/Eurostat/WHO 2017), the health component of LTC spending relates to nursing care and personal care services (help with ADLs). It also covers palliative care and care provided in LTC institutions (including costs for room and board) or at home. LTC social expenditure primarily covers help with IADLs. Table 1 provides data summary statistics.

From Table 1, it can be seen that, on average, the number of “recipients” at home is three times the number of “recipients” in institutions. Hence, the importance of including this second output in the analysis stands out. On average, there are 1.16 beds in institutions per recipients and 2.69 recipients (in institutions or at home) per worker.

The literature provides empirical guidelines regarding the appropriate balance between the number of decision-making units (DMUs) and the number of variables in efficiency analysis (Golany and Roll 1989; Homburg 2001; Nunamaker 1985; Banker et al. 1989; Friedman and Sinuany-Stern 1997; Raab and Lichty 2002; Dyson et al. 2001). We follow the criteria proposed by Cooper et al. (2007), stating that DMUs should be greater than or equal to the maximum of the product of inputs and outputs or three times the sum of inputs and outputs, expressed as DMUs ≥ max(Input × Output, 3(Input + Output)). In our case, we satisfy the stringent restriction because 16 ≥ max (3 × 2, 3(3 + 2)) = 15.

Finally, we analyze the incidence of government structure in the productivity of LTC provision. We explore whether unitary or federal government structures increase their productivity at different paces or not. The organization of the government can shed light on efficiency and productivity in service provision since federal countries would tend to have a more decentralized provision.

Considering the governance structure, the Center for the Study of Federalism (CSF 2023) classifies Austria, Canada, Germany, Spain, Switzerland, and the United States as federal systems, characterized by the consensus on the autonomy of political, administrative and jurisdictional powers among central and regional authorities. On the other hand, Denmark, Estonia, Hungary, Israel, Korea, Luxembourg, Netherlands, Norway, Slovakia, and Sweden as unitary forms of government are based on a political unicity superimposed on an administrative hierarchical structure.

In

Table 2. Public expenditure per capita by level of government (USD PPP 2016).

	Public Expenditure per Capita (USD PPP)				Share of Total Expenditure (%)
Unitary States	Local	State	Central	Total	Local	State	Central
Denmark	17,070	-	9203	26,273	65%	0%	35%
Estonia	2797	-	9285	12,082	23%	0%	77%
Hungary	1609	-	10,856	12,465	13%	0%	87%
Israel	2063	-	12,676	14,739	14%	0%	86%
Korea	5089	-	6731	11,820	43%	0%	57%
Luxembourg	5033	-	37,941	42,974	12%	0%	88%
Netherlands	6961	-	14,994	21,955	32%	0%	68%
Norway	9915	-	19,966	29,881	33%	0%	67%
Slovak Republic	1998	-	10,645	12,643	16%	0%	84%
Sweden	12,238	-	11,947	24,185	51%	0%	49%
Federal States
Austria	4298	4876	16,433	25,607	17%	19%	64%
Canada	3861	10,283	4428	18,572	21%	55%	24%
Germany	3900	6494	11,223	21,617	18%	30%	52%
Spain	2095	5449	7782	15,326	14%	36%	51%
Switzerland	4745	8722	8464	21,931	22%	40%	39%
United States	10,533		11,247	21,780	48%		52%

Source: OECD Regions and Cities at a Glance 2018: Public expenditure per capita by level of government (USD PPP 2016). In OECD (2018).

, we present public expenditure per capita by level of government and its share into total expenditure for unitary and federal countries. On average, 70% of total public expenditure is in the hands of the central government for unitary countries, while that number shrinks to 47% for federal countries.

It is important to note that (i) federalism is not the same as decentralization: federalism is a constitutional decision, while decentralization arises from a negotiation process between the different levels of governments; and (ii) unitary states can embrace significant levels of administrative and financial decentralization through local governments. Table 2 shows that some unitary governments transfer a high spending capacity to local governments (Denmark 65% and Sweden 51%) and, with it, a wide range of competences. These data confirm that constitutional definitions of the distribution of political competences between federal and unitary states do not always correspond in practice to an effective distribution of power, because decentralization without financial resources is inoperative.

Great efforts are currently being made to try to understand and evaluate the degree of decentralization in OECD countries in terms of the responsibilities assigned in obtaining resources, their expenditure, transfers between regions, and coordination between different levels of government (De Biase and Dougherty 2023).

To illustrate the difference between unitary government and decentralization capacity, De Biase and Dougherty (2023) reveal that more than 90% and 70% of LTC expenditures are decentralized to local governments in Denmark and Sweden, respectively. In contrast, in other unitary countries such as Norway, Estonia, the Netherlands, Israel, Slovakia, Luxembourg, and Hungary, spending levels in local governments range from 25% in the former to barely 1% in the latter, with central governments maintaining the highest level of spending. In the case of Spain, as a federal state, 70% of LTC spending is by regional governments (Autonomous Communities). Unfortunately, these authors do not provide information for all the countries in our study to explore those determinants.

5. Results

In an output-oriented model, the efficiency scores depict the percentage of potential product (or output) that is actually produced.

We choose the output orientation because, although all inhabitants are eligible, as noted in Section 2, instances of rationing have been detected where more expensive substitutes (hospitals) are used or where rationing occurs due to regulations. Therefore, we analyze how countries could be more efficient in providing more LTC services with the available resources. On the contrary, an input orientation could lead to reductions in personnel and, consequently, in quality, or alternatively, increases in waiting lists (Norton 2000; Peña-Longobardo et al. 2016; Lopes et al. 2018; Albuquerque 2022).

In this section, we analyze the technical efficiency scores and then we delve into the Malmquist Productivity Index to analyze changes in efficiency.

5.1. Technical Efficiency

In

Table 3. Technical efficiency scores by year ($D_v\left(X_o,Y_o\right)$).

Country	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019	Mean	Year Eff.
Austria	0.731	0.688	0.801	0.789	0.790	0.786	0.776	0.761	0.883	0.903	0.791	0
Canada	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Denmark	0.838	0.766	0.836	0.794	0.787	0.772	0.765	0.846	0.814	0.830	0.805	0
Estonia	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Germany	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Hungary	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Israel	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Korea	0.985	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.998	9
Luxembourg	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Netherlands	1.000	1.000	1.000	1.000	1.000	0.889	0.870	0.838	0.827	0.845	0.927	5
Norway	0.801	0.762	0.982	0.925	0.938	0.945	0.944	0.907	0.877	0.910	0.899	0
Slovak Republic	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Spain	0.767	0.743	0.729	0.686	0.854	0.899	0.957	0.895	0.927	0.935	0.839	0
Sweden	0.874	0.659	0.834	0.873	0.869	0.865	0.848	0.757	0.753	0.754	0.809	0
Switzerland	0.870	0.845	0.913	0.907	0.910	0.922	0.942	0.913	0.912	0.913	0.905	0
United States	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	10
Mean	0.929	0.904	0.943	0.936	0.947	0.942	0.944	0.932	0.937	0.943	0.936
Minimum	0.731	0.659	0.729	0.686	0.787	0.772	0.765	0.757	0.753	0.754	0.659
Maximum	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
Standard Dev.	0.098	0.134	0.091	0.100	0.079	0.079	0.083	0.089	0.083	0.078	0.083

, we calculate the technical efficiency scores of 16 countries performing an output-oriented DEA under the assumption of Variable Returns to Scale.

Table 3 presents the technical efficiency scores of the 16 OECD countries and their distribution over the years. The last column (Year Eff.) shows how many times a certain country was on the efficient frontier. The mean technical efficiency for the whole sample is 0.936, ranging from 0.659 to 1.00. Throughout the decade, there is not a clear pattern indicating an increase or decrease in relative efficiency. Looking at the distribution of efficiency scores of countries by years, we see that the standard deviation shrinks in the last few years.

In Figure 2, we depict technical efficiency by country. We can observe two groups of countries. The group on the left includes the least efficient countries (Austria, Denmark, Sweden, Spain, Norway, Switzerland and the Netherlands). These countries have more variable technical efficiency scores. On the contrary, the group on the right includes the most efficient countries that are less variable in their scores (Korea, Luxembourg, Estonia, Canada, Hungary, Slovakia, United States, Israel, and Germany).

In order to determine confidence intervals for the efficiency scores, we applied the Simar and Wilson (1998) algorithm. This algorithm works by resampling the data with replacements to create a bootstrap sample. Then, a new DEA model is estimated using the bootstrap sample, and the efficiency scores for each DMU are calculated. This process is repeated a large number of times to generate a distribution of bootstrap efficiency scores for each DMU. The sampling distribution of the DEA efficiency scores can then be used to estimate confidence intervals, conduct hypothesis tests, and calculate bootstrap p-values. In our case, we set 2000 iterations, using “bootstrap_basic” from the deaR package 1.4.1 in R (Coll-Serrano et al. 2023), for the first and the last year in our sample. The application of the bootstrap method to the DEA scores introduces a small degree of variability, resulting in efficiency scores slightly below 1 for some countries initially classified as efficient. This method captures the uncertainty in efficiency estimates. The confidence intervals derived from the bootstrapping process provide a more robust understanding of the efficiency scores, allowing us to account for potential noise in the data.

In Figure 3, after applying the technique to obtain confidence intervals, we can distinguish between the efficient/inefficient countries. The first group (United States, Slovakia, Luxembourg, Israel, Estonia, Canada, Germany, Hungary, and Korea) can be considered efficient; the inefficient group has confidence intervals smaller than the efficient group. Sweden, Austria, Switzerland, Norway, and Spain had a score of 0.90 and could not be considered efficient in any case. Compared with 2011, we can see that all inefficient countries (except the Netherlands and Korea) improved their technical efficiency, and this improvement is statistically significant.

Finally, in

Table A1. Intensity variables (${\lambda }_j$) by year.

	Country	Canada	Estonia	Germany	Hungary	Israel	Korea	Luxem	Neth	Slovak. Repc	United. States
2010	Austria	0.049	-	-	-	0.031		-	0.125	0.796	-
2011	Austria	-	-	-	0.124	-	-	-	0.169	0.707	-
2012	Austria	-	-	-	-	-	-	0.404	0.209	0.387	-
2013	Austria	-	0.092	-	-	-	0.312	-	-	0.595	-
2014	Austria	0.011	0.140	-	-	-	0.264	-	-	0.585	-
2015	Austria	-	0.152	-	-	-	0.257	-		0.590	-
2016	Austria	-	0.132	-	-	-	0.239	-		0.629	-
2017	Austria	0.028	-	-	-	-	0.207	-		0.766	-
2018	Austria	-	0.308	-	0.535	-	0.157	-		-	-
2019	Austria	-	0.310	-	0.554	-	0.137	-		-	-
Average	Austria	0.009	0.113	-	0.121	0.003	0.175	0.040	0.100	0.505	-
2010	Denmark	0.051	-	-	-	0.581		-	0.078	0.290	-
2011	Denmark	-	-	-	0.139	0.536	-	-	0.108	0.217	-
2012	Denmark	-	-	-	-	0.438	-	0.312	0.147	0.104	-
2013	Denmark	0.066	0.415	-	-	0.374	0.145	-	-	-	-
2014	Denmark	0.054	0.449	-	-	0.339	0.158	-	-	-	-
2015	Denmark	0.047	0.466	-	-	0.326	0.162	-		-	-
2016	Denmark	-	0.202	-	-	0.349	0.184	-		0.265	-
2017	Denmark	-	0.528	-	-	0.286	0.185	-		-	-
2018	Denmark	-	0.554	-	-	0.268	0.178	-		-	-
2019	Denmark	-	0.619	-	-	0.219	0.162	-		-	-
Average	Denmark	0.022	0.323	-	0.014	0.372	0.130	0.031	0.066	0.088	-
2010	Korea	-	-	-	-	-	-	-	0.113	0.844	0.043
Average	Korea	-	-	-	-	-	-	-	0.113	0.844	0.043
2015	Neth	0.385	-	0.037	-	-	0.577	-	-	-	-
2016	Neth	0.272	-	0.025	-	-	0.703	-	-	-	-
2017	Neth	0.387	-	-	-	-	0.602	-	-	0.012	-
2018	Neth	0.382	-	-	0.031	-	0.588	-	-	-	-
2019	Neth	0.384	-	-	0.093	-	0.522	-	-	-	-
Average	Neth	0.362	-	0.012	0.025	-	0.598	-	-	0.002	-
2010	Norway	-	-	-	-	0.550		0.277	0.174	-	-
2011	Norway	-	-	-	-	0.480	-	0.341	0.179	-	-
2012	Norway	-	-	-	-	0.352	-	0.511	0.137	-	-
2013	Norway	-	0.366	-	-	0.424	0.210	-	-	-	-
2014	Norway	-	0.410	-	-	0.397	0.192	-	-	-	-
2015	Norway	-	0.427	-	-	0.397	0.175	-		-	-
2016	Norway	-	0.453	-	-	0.381	0.166	-		-	-
2017	Norway	-	0.461	-	-	0.376	0.164	-		-	-
2018	Norway	-	0.476	-	-	0.368	0.155	-		-	-
2019	Norway	-	0.575	-	-	0.282	0.143	-		-	-
Average	Norway	-	0.317	-	-	0.401	0.134	0.113	0.098	-	-
2010	Spain	0.097	-	0.060	-	0.750		-	-	-	0.093
2011	Spain	-	-	0.265	0.688	-	-	-	-	-	0.047
2012	Spain	-	-	0.260	0.697	-	-	-	-	-	0.042
2013	Spain	-	-	0.235	0.717	-	-	-	-	-	0.049
2014	Spain	-	-	0.222	0.727	-	-	-	-	-	0.051
2015	Spain	-	-	0.234	0.716	-	-	-		-	0.050
2016	Spain	-	-	0.228	0.715	-	-	-		-	0.057
2017	Spain	-	-	0.205	0.726	-	-	-		-	0.070
2018	Spain	-	-	0.248	-	-	-	-		0.689	0.064
2019	Spain	-	-	0.241	-	-	-	-		0.688	0.071
Average	Spain	0.010	-	0.220	0.499	0.075	-	-	-	0.138	0.059
2010	Sweden	-	-	-	-	0.098		-	0.195	0.661	0.046
2011	Sweden	0.053	-	-	-	0.353	-	-	0.589	-	0.006
2012	Sweden	-	-	-	-	0.354	-	0.141	0.505	-	-
2013	Sweden	0.137	-	-	-	0.251	0.612	-	-	-	-
2014	Sweden	0.144	-	-	-	0.343	0.514	-	-	-	-
2015	Sweden	0.136	-	-	-	0.364	0.501	-		-	-
2016	Sweden	0.036	-	-	-	0.319	0.645	-		-	-
2017	Sweden	-	-	-	-	0.359	0.625	-		-	0.016
2018	Sweden	-	-	0.026	-	0.398	0.571	-		-	0.004
2019	Sweden	0.042	0.103	-	-	0.235	0.620	-		-	-
Average	Sweden	0.055	0.010	0.003	-	0.307	0.454	0.014	0.258	0.066	0.007
2010	Switz	0.207	-	-	-	0.267		-	0.114	0.412	-
2011	Switz	0.095	-	-	0.445	0.314	-	-	0.147	-	-
2012	Switz	-	-	-	0.344	0.312	-	-	0.229	0.115	-
2013	Switz	0.211	0.332	-	-	0.244	0.212	-	-	-	-
2014	Switz	0.190	0.360	-	-	0.216	0.235	-	-	-	-
2015	Switz	0.223	0.312	-	-	0.314	0.152	-		-	-
2016	Switz	0.170	-	-	-	0.490	0.125	-		0.215	-
2017	Switz	0.174	-	-	-	0.485	0.129	-		0.212	-
2018	Switz	0.197	0.241	-	-	0.444	0.118	-		-	-
2019	Switz	0.190	0.275	-	-	0.409	0.125	-		-	-
Average	Switz	0.166	0.152	-	0.079	0.350	0.122	-	0.098	0.095	-

in Appendix A we present the values of the intensity variables (${\lambda }_j$) by year for each country. The efficient countries are presented in the columns, while the inefficient countries are presented in the rows. The sum of all the lambdas horizontally per year is equal to 1. In this way, if we read the rows, we see for each inefficient country which should be their reference countries to improve efficiency; for example, Austria has an inefficient provision and, in order to improve it, it should use Slovakia, Hungary, and Korea as references. If we look at the columns, we can observe which inefficient countries take each efficient country as a reference; for example, the United States is an efficient reference for Korea, Spain, and Sweden.

If an inefficient country assigns a lambda weight greater than zero to a country on the frontier, this suggests that the latter is a reference. The magnitude of lambda indicates how important this reference is. For example, if the Netherlands assigns a value of 0.598 to South Korea’s lambda, this means that the Netherlands should copy 60% of Korea’s output and input mix and 40% of those of other countries, since the model used is the BCC and all lambdas must add up to one.

In Austria, the average efficiency is 0.791. The main referents for Austria are Slovakia for the first eight years and Hungary for the last two. Denmark and Norway show an average efficiency of 0.805 and 0.899, respectively. Denmark and Norway should learn from Israel and, in a second instance, from Estonia, although Israel’s incidence has been falling over time while Estonia’s has been increasing. The Netherlands has an efficiency of 0.927 but this is an average of 1 in the first five years in which that country was efficient and 0.854 in the following years; in the last part of the decade, the Netherlands should learn from Korea and Canada. Sweden has an efficiency level of 0.809 and should learn from Israel and Korea, but mainly Korea. All the Nordic countries in our study (Denmark, Norway, and Sweden) should learn from Israel. It is consistent, as the Scandinavian countries tend to provide more services at home; Israel has seen an increase of over 72% of residents at home in the last decade. Switzerland has an efficiency level of 0.905. Like the Scandinavian countries, Switzerland maintains recipients in institutions (Y1) and significantly increases recipients at home (Y2); this is what Israel achieves. Spain has an average efficiency of 0.839, offering continuous and stable behavior in the growth of all outputs and inputs in the 2010 to 2019 period. Spain should learn from Hungary for the first years (until 2017) and from Slovakia afterwards (2018 and 2019). In a second instance, it should learn from Germany (0.220).

In

Table 4. Technical efficiency scores by government structure.

Government Structure	N	Mean	St. Dev	Min	Max
Unitary	100	0.944	0.087	0.659	1.000
Federal	60	0.923	0.097	0.686	1.000

, we retrieve the distinction between unitary and federal government structure, and present the average of the efficiencies for the two categories. Although the average efficiency for governments with a unitary structure is higher than that for countries with a federal government structure, statistically the averages are not different. On the other hand, from Figure 4 it is observed that the efficiency of governments with a federal structure went from 0.90 in 2010 to 0.96 in 2019, while in that period the average efficiency of countries with a unitary structure went from 0.95 in 2010 to 0.94 in 2019.

To statistically analyze the difference between the growth rates of efficiency in federal and unitary countries, given the small sample size, a Tobit model (Tobin 1958) was estimated and is shown in

Table 5. Random-effects Tobit regression—dependent variable technical efficiency score ($D_v\left({\mathrm{X}}_{\mathrm{o}},{\mathrm{Y}}_{\mathrm{o}}\right)$) for 16 countries in 2010–2019.

Efficiency	Coef.	St.Err.	t-Value	p-Value	[95% Conf	Interval]	Sig
Federal	−0.112	0.093	−1.20	0.23	−0.294	0.07
Federal*trend	0.017	0.004	3.87	0	0.009	0.026	***
HDI	−0.545	0.797	−0.68	0.494	−2.106	1.017
Constant	1.615	0.71	2.28	0.023	0.225	3.006	**
sigma_u	0.228	0.063	3.61	0	0.104	0.352	***
sigma_e	0.062	0.006	10.79	0	0.051	0.074	***
Mean dependent var		0.936	SD dependent var			0.091
Number of obs		160	Chi-square			17.499
Prob > chi2		0.001	Akaike crit. (AIC)			−101.278
LR test of ${\sigma }_v$ = 0 chibar2(01) =		169.86	Prob ≥chibar2 =			0.000

*** p < 0.01, ** p < 0.05.

.

In the second stage of the analysis, a Tobit model (Tobin 1958) is employed to regress the DEA-derived efficiency scores on a set of explanatory variables, including government structure, allowing us to assess the impact of institutional and demographic factors on efficiency levels across the sampled countries. Tobin (1958) models the relationship between independent variables and a censored dependent variable, where values are constrained by an upper or lower limit (or both). The Tobit regression estimates the effects of explanatory variables on the latent (uncensored) variable while accounting for the fact that observed values cannot exceed the specified threshold. This is particularly useful in cases like efficiency scores, in which maximum efficiency values are truncated at 1, ensuring more accurate and unbiased estimates than would be obtained from a standard linear regression. Since the data have a panel structure, we used the random effects version to control for possible unobserved heterogeneity (Naylor and Smith 1982; Pendergast et al. 1996; Skrondal and Rabe-Hesketh 2004).

In this model, efficiency is explained by a dummy variable (Federal) that captures whether the country is federal, an interaction (Federal*trend) that captures if the evolution of efficiency over time differs between unitary and federal countries, and a control variable (HDI)1. The Human Development Index (HDI) is an index annually produced by the Human Development Report Office of the United Nations Development Programme (UNDP) (see https://hdr.undp.org/data-center/human-development-index#/indicies/HDI accessed on 10 July 2023). This index is a summary measure of average achievement in key dimensions of human development: a long and healthy life (Life Expectancy Index), being knowledgeable (Education Index), and having a decent standard of living (GNI Index). This last variable jointly controls three types of effects on efficiency: (i) that greater life expectancy implies a longer time spent living in unhealthy conditions, (ii) that greater education implies demanding higher-quality services, and (iii) that higher levels of income imply more expensive services (Baumol effect).

The equation estimated is as follows:

$${Efficiency Score}_{jt}^{*}={\beta }_0+{\beta }_1.{Federal}_j+{\beta }_2.{\left(Federal*Trend\right)}_j{+\beta }_3.{HDI}_{jt}+v_j+{\epsilon }_{jt}$$

(5)

where the random effects $v_j$ are i.i.d, $N(0,{\sigma }_v^2)$, and ${\epsilon }_{jt}$ is the error term and is assumed to follow a normal distribution with a mean of zero and constant variance ${\epsilon }_{jt}~N(0,{\sigma }_{\epsilon }^2)$ independently of $v_j$.2

$${Efficiency Score}_{jt}\left\{\begin{array}{c}{Efficiency Score}_{jt}^{*} if { Efficiency Score}_{jt}^{*}\le 1 \\ 1 if { Efficiency Score}_{jt}^{*}>1\end{array}\right.$$

(6)

From the regression in

Table 6. Simar and Wilson (2007)—dependent variable technical efficiency score ($D_v\left({\mathrm{X}}_{\mathrm{o}},{\mathrm{Y}}_{\mathrm{o}}\right)$) for 16 countries in 2010–2019.

Efficiency	Observed Coef.	Bootstrap Std. Err	z	p > z	Percentile 95% Conf.	Interval
Federal	−0.034	0.044	−0.750	0.451	−0.119	0.057
Federal*trend	0.021	0.009	2.490	0.013	0.005	0.039
HDI	−2.916	0.574	−5.080	0.000	−4.108	−1.867
_cons	3.491	0.532	6.560	0.000	2.535	4.590
$\sigma$	0.142	0.013	10.800	0.000	0.116	0.167

, it is observed that the efficiency in federal countries is lower than that in unitary countries. However, over time, federal countries have improved their efficiency, while unitary countries have not. Finally, the negative sign of the HDI indicates that the higher the longevity, education level, or income level of the population, the lower the efficiency. This last result is consistent with Serrano-Alarcón et al. (2021), Siciliani (2013), and (Ozbugday et al. 2020). Finally, the likelihood-ratio test comparing the random effects model with the pooled model (Tobin 1958) rejects the null hypothesis, indicating the presence of significant individual effects.

The Tobit model assumes a normally distributed error term, which may not hold in our context, given that efficiency models are truncated at the upper bound of 1. Therefore, to estimate confidence intervals that adequately capture the variability of efficiency scores and address the limitations imposed by this truncation, we apply the Simar and Wilson (2007) methodology. This approach enables the generation of robust confidence intervals through resampling techniques, providing more reliable estimates of the effects of the explanatory variables on efficiency. The results, presented in Table 6, show that federal countries do not exhibit a statistically lower level of technical efficiency compared to unitary countries, a finding consistent with the Tobit model. However, the improvement in technical efficiency for federal countries surpasses that of unitary countries, and this difference is statistically significant in both models. Additionally, the Human Development Index (HDI) retains the negative sign observed in the Tobit model and is statistically significant in this case. Combining both models, we observe that increases in the HDI are associated with a deterioration in efficiency, although this relationship weakens when individual effects are considered.

The policy implications of these results imply that it may be beneficial for federal countries to continue with structural reforms that promote decentralization and local autonomy. Unitary countries, on the other hand, should critically evaluate their current policies and consider adopting effective practices from federal countries.

To evaluate the possible sources of efficiency improvement over time, in Section 5.2, we decompose the productivity changes using the Malmquist Index by grouping the countries into unitary and federal groups.

5.2. Malmquist Productivity Index

It is worth mentioning that technical efficiency is a static concept; it establishes the distance of every country to the best practice frontier of that year. In order to analyze whether a country is improving its efficiency, how the efficient frontier moves over time and the reasons why it does so, we estimate the Malmquist Productivity Index ($M_c$).

In

Table 7. Malmquist Productivity Index and its components by year.

Period	${\mathit{M}}_{\mathit{c}}$	TEC	TC	SEC
2010–2011	0.981	0.967	1.033	0.982
2011–2012	0.988	1.050	0.876	1.074
2012–2013	1.034	0.991	1.102	0.947
2013–2014	1.000	1.014	1.016	0.971
2014–2015	0.988	0.995	0.982	1.010
2015–2016	1.003	1.001	1.010	0.992
2016–2017	1.016	0.987	1.005	1.024
2017–2018	1.020	1.006	0.983	1.032
2018–2019	1.014	1.007	0.999	1.008
Average	1.005	1.002	0.999	1.004

Technical efficiency change (TEC), technological efficiency change (TC), and change in scale efficiency (SEC).

, we present the average Malmquist Productivity Index ($M_c$) and its decomposition for each year.3 On average, there was an increase of 0.5%. However, in the first part of the decade, the index tended to decrease and in the second part of the decade it tended to increase. In addition, after 2015, all the variables present a lower variance. For the first part of the decade, our results are consistent with the productivity change throughout the 2010–2014 period calculated by Ozbugday et al. (2020), which is a 0.1% increase in the productivity of LTC provision.

The 0.5% increase in the productivity index could mitigate part of the pressure on the public budget for financing the service. Nonetheless, that improvement might not be enough. Recall that for OECD countries, an average two- or three-fold increase in their ratios of public LTC spending to GDP by 2060 is expected.

The last three columns decompose the $M_c$ in, $TEC$, $TC$, and $SEC$. In no case were the three indexes above or below 1; this means, for example, that technological change was compensated by technical change or scale efficiency change.

On the other hand, taking only the changes in technical efficiency and in technology, it is observed that only in two of the nine periods did both values grow simultaneously. In the other seven remaining cases, when innovation increased (technological efficiency change was above 1), technical efficiency change decreased (was below 1), and when technical efficiency increased, innovation did not increase.

The last row shows that, on average, the increase of 0.5% in productivity is mainly explained by the increase in the scale efficiency (0.4%) and the technical efficiency change (0.2%). This could be evidence that the inefficient countries are improving their productivity at a pace that doubles that of the more efficient countries (0.1%) and, in the long run, we could expect a convergence in the technology of provision.

Next, in

Table 8. Malmquist Productivity Index and its components by government structure.

Period	${\mathit{M}}_{\mathit{c}}$			TC			TEC			SEC
	2010–2015	2015–2019	2010–2019	2010–2015	2015–2019	2010–2019	2010–2015	2015–2019	2010–2019	2010–2015	2015–2019	2010–2019
Netherlands	0.921	0.988	0.954	0.999	0.979	0.989	0.998	0.966	0.982	0.971	1.041	1.005
Sweden	0.968	0.985	0.976	0.985	0.974	0.980	0.977	0.987	0.982	0.957	1.027	0.992
Luxembourg	0.932	1.035	0.982	1.002	0.992	0.997	1.000	1.000	1.000	0.930	1.043	0.985
Slovak Rep	1.011	0.979	0.995	1.011	0.979	0.995	1.000	1.000	1.000	1.000	1.000	1.000
Denmark	0.968	1.024	0.996	1.000	1.002	1.001	0.984	1.019	1.001	0.984	1.004	0.994
Hungary	1.003	0.995	0.999	0.983	0.995	0.989	1.000	1.000	1.000	1.020	1.000	1.010
Norway	0.999	1.002	1.001	0.996	0.993	0.995	1.034	0.991	1.012	0.970	1.018	0.994
Korea	1.019	1.009	1.014	0.996	1.008	1.002	1.003	1.000	1.002	1.020	1.001	1.011
Estonia	1.082	0.974	1.027	1.007	0.974	0.990	1.000	1.000	1.000	1.075	1.000	1.037
Israel	1.016	1.046	1.031	1.016	1.046	1.031	1.000	1.000	1.000	1.000	1.000	1.000
Mean Unitary	0.991	1.003	0.997	0.999	0.994	0.997	0.999	0.996	0.998	0.992	1.013	1.003
Austria	0.995	1.018	1.007	1.003	0.985	0.994	1.015	1.035	1.025	0.978	0.999	0.988
Switzerland	1.007	1.010	1.009	0.999	1.006	1.003	1.012	0.998	1.005	0.996	1.007	1.001
Canada	1.009	1.014	1.011	0.994	1.008	1.001	1.000	1.000	1.000	1.015	1.006	1.011
Germany	1.000	1.051	1.025	0.998	1.013	1.005	1.000	1.000	1.000	1.002	1.037	1.019
United States	1.036	1.018	1.027	1.008	1.018	1.013	1.000	1.000	1.000	1.028	1.000	1.014
Spain	1.013	1.072	1.042	0.986	1.019	1.002	1.032	1.010	1.021	0.995	1.042	1.018
Mean Federal	1.014	1.038	1.026	0.996	1.014	1.005	1.008	1.002	1.005	1.010	1.021	1.016
M–W U test	654	509	1.865	788	660	2340	735.5	585	2197.5	692	719	2333
p-value	0.3426	0.0168	0.0198	0.7094	0.3738	0.7119	0.8751	0.0714	0.2849	0.5651	0.7572	0.687

Technical efficiency change (TEC), technological efficiency change (TC), and change in scale efficiency (SEC).

we present the average of the Malmquist Productivity Index for three different periods: (i) 2010–2015, (ii) 2015–2019, and (iii) 2010–2019. In turn, to facilitate the analysis we have separated the countries into two groups according to the government structure and we have ordered each group in ascending order based on the Malmquist Productivity Index for the 2010–2019 period (Column 3).

It can be seen that for countries with a unitary structure, productivity on average has not increased (0.997). Distinguishing the periods, from 2010 to 2015 there was a drop in productivity (0.991) and from the period 2015 to 2019 there was an increase (1.003). Decomposing the Malmquist Index, it is observed that the increase registered in 2015 to 2019 is exclusively due to an improvement in the efficient scale of provision (1.013) since the change in technological efficiency is 0.997 and the change in technical efficiency is 0.998.

In the case of countries with a federal structure, an average productivity increase of 2.6% is observed. This change in productivity was greater in the second period (average 3.8%) than in the first period (average 1.4%). Contrary to what is described in the case of unitary countries, when the Malmquist Index is decomposed, it is observed that the technical, technological, and scale of provision changes are positive in all sub-periods with the exception of the change in technological efficiency in the 2010–2015 period (0.996).

The Mann–Whitney U test Mann and Whitney (1947) is presented at the bottom of Table 8 to statistically assess differences in productivity changes between unitary and federal countries. The results indicate that productivity varied over the 2010–2019 period, primarily driven by the productivity gap observed in the second sub-period, 2015–2019. Further decomposition reveals that this differential is largely explained by variations in TEC.

In both sub-periods, the improvement in scale efficiency stands out. Then, the technical efficiency has remained relatively constant at 1.008 in the first part and 1.002 in the second, with the change in technological efficiency rising from 0.996 to 1.014 in the second part.

One of the reasons why the productivity of federal countries can be higher than that of unitary countries could be the political and fiscal accountability that decentralization exercises and the possibility of competition between different jurisdictions. At some point it works as a yardstick competition (Shleifer 1985), in which the more direct the comparison between different units, the easier it is to evaluate the quality of the decisions made. In this sense, economists have begun to see decentralization as a way to adjust principal–agent problems between constituents and incumbents (Besley 2007).

6. Conclusions

In this article, we estimated efficiency and productivity changes in the LTC sector in OECD countries over the 2010–2019 period.

Many developed countries are facing challenges with the fiscal sustainability of LTC spending. While conditions, norms, and preferences may differ between countries, it is universally expected that individuals will need to contribute more to the financing of LTC services compared to other healthcare services. The increasing demand and decreasing supply of LTC services have led most OECD countries to alter their current structures and financing schemes to ensure that their LTC systems are affordable, sustainable, efficient, and equitable. However, despite the growing concern over the rising costs of LTC, the issue of efficiency and productivity in LTC provision has received little attention in the literature.

The economic literature on LTC has largely focused on understanding the factors contributing to the growth of LTC costs, optimizing the organization of LTC services, unraveling the complexities of LTC insurance, exploring ownership structures, analyzing government regulations, assessing competition dynamics, and delving into the dynamics of informal care. While these are unquestionably vital aspects, it is equally imperative to evaluate and analyze the performance of countries in delivering LTC services. The reason is that productivity enhancements have the potential to counterbalance future cost escalations. In this paper, we undertook an examination of the potential productivity gains by estimating the efficiency levels in the provision of LTC across OECD countries. We applied a non-parametric approach, specifically, DEA efficiency analysis output orientation and the Malmquist Productivity Index.

The results of our study reveal that the OECD countries within the sample exhibited an average efficiency level of 94% and experienced a modest 0.5% increase in the productivity of LTC provision between 2010 and 2019. This estimated level of efficiency probably falls short of adequately addressing the fiscal pressures stemming from the growing number of LTC beneficiaries and the associated LTC expenditure. Despite the substantial rise in both the number of LTC beneficiaries and the overall LTC spending, the efficiency in LTC provision might not be sufficiently effective in mitigating the fiscal challenges. This underscores the pressing need for substantial improvements in the efficiency of LTC provision across OECD countries.

We identified the best performing countries, the technically efficient ones, and provided referent countries for the inefficient ones. In considering the political organization, we found that over the years, the average efficiency of unitary countries was relatively the same but the average efficiency of federal countries was lower at the beginning and higher at the end. We statistically proved that federal countries improve their efficiency over time while unitary countries do not. So, moving to the intertemporal ground, in unitary countries there was no significant increase in productivity over the years, with a slight drop from 2010 to 2015 and a modest increase from 2015 to 2019, mainly due to an improved and efficient scale of provision. In contrast, federal countries saw an average productivity increase of 2.6%, with a more significant increase from 2015 to 2019. The decomposition of the Malmquist Index showed positive changes in technical, technological, and scale of provision in federal countries, except for a slight drop in technological efficiency from 2010 to 2015. The higher productivity in federal countries may be attributed to decentralization, fostering political and fiscal accountability and competition among jurisdictions, which can help address principal–agent problems.

While the results of the DEA provide insight into the efficiency levels of LTC provision in OECD countries, there is still a need to conduct further research on the determinants of such efficiency and productivity levels. Understanding the factors that contribute to the inefficiencies in LTC provision can help policymakers develop strategies to improve the quality and accessibility of LTC services while ensuring the fiscal sustainability of LTC spending. Possible factors that can be explored include the organizational structure of LTC providers, the availability and quality of human resources, the level of technological innovation, and the regulatory framework. Further research can also shed light on the potential impact of these factors on the cost-effectiveness of LTC provision and ultimately improve the well-being of LTC recipients and their families.