The cost of readmissions in hospitals: the case of the Spanish public hospitals

Abstract

Background

In this paper, we propose a novel model that allows us to understand the effect of hospital readmissions on technology and costs. To do this, we consider that hospitals may experience heterogeneous discharges: on the one hand, discharges corresponding to patients who have completed their healing process in hospital and, on the other hand, discharges resulting from patients who have been discharged too early and are therefore required to be readmitted to hospital. In the first case, discharges involve more resources; in the second case, the patient returns implying an additional use of resources. In tandem, two new issues arise which need to be addressed: a) Does a trade-off exist between the decision to discharge at the finalisation of fully completed treatment or the decision to discharge taken at an earlier stage; b) Readmissions may prove endogenous and if so, their econometric treatment must be considered in order to obtain unbiased results. Our study contributes to the literature by proposing a novel model which estimates the marginal cost of readmissions, thus allowing us to understand the effect of readmission on technology and hospital costs.

Methods

To resolve the foregoing concerns, this paper proposes a theoretical and empirical model based on the dual theory, which combines cost and input-oriented distance functions to obtain the marginal cost of readmissions. Our empirical application uses a panel of Spanish public hospitals observed over the period 2002–2016.

Results

Results indicate that the treatment required by a patient who is readmitted proves more expensive than keeping the patient under observation for a few extra days in order to achieve a definitive discharge. Moreover, this additional cost follows an increasing temporal trend, especially in times of expansion when the availability of resources is greater.

Conclusions

Given that the results indicate that readmissions imply an additional cost for the hospital system, they must be contained. In fact, readmission rates are a significant component of current hospital sector activity improvement strategies. Therefore, knowing the cost which readmission implies is relevant for the design of policies that seek to penalize those hospitals with high readmission rates.

Background

For most European countries, the public provision of health care is one of the main pillars of the Welfare State. However, over recent years this sector faces significant challenges due to the need to increase innovation in medicines and treatments. To this, we must add an ageing population which increases morbidity, and restrictive fiscal policies that have had a negative impact on healthcare budgets, especially in countries with tax-financed healthcare systems [1].

Due to the foregoing scenario, reforms are required that improve the efficiency of the healthcare system and reduce any unnecessary costs which would threaten the sustainability of said system. As [2] point out, although these reforms are not only applicable to the hospital sector, the latter is particularly affected by the aforementioned problems, given that it represents a significant portion of total healthcare expenditure worldwide.

In this context, policies aimed at improving hospital activity are becoming increasingly important. Among them are those which monitor and/or penalize unnecessary hospital readmissions in order to increase the quality of hospital care and reduce costs [3].

However, to understand the relationship between readmissions and hospital cost, it is necessary to take into account that, given the existence of hospital readmissions, hospitals have heterogeneous discharges, with two scenarios being contemplated: on the one hand, discharges corresponding to those patients who have completed their healing process in the hospital and, on the other hand, the discharges resulting from patients who have been discharged too early and therefore require readmission to the hospital.

In the first case, discharges involve more resources; in the second case, the patient is readmitted with the subsequent additional use of resources. In this way, a new issue arises which is the main question of this study: Does a trade-off exist between the decision to discharge at the finalization of fully completed treatment or the decision to discharge taken at an earlier stage? Which of the two options is more costly?

To the best of our knowledge, this is the first paper that aims to answer this question. Thus, our study contributes to the literature by proposing a novel model, based on the dual theory between the distance and cost functions, that allows us to understand the effect of readmission on hospital costs, by estimating the marginal cost of readmissions. In addition, we take into account that readmissions may prove endogenous and if so, their econometric treatment must be considered in order to obtain unbiased results.

Finally, the study allows an analysis of other key characteristics of technology, as well as providing a series of useful productive efficiency indices for hospitals via an examination of the determinants of said efficiency.

The results indicate that readmissions add extra costs to the hospital system, making it essential to control them. Therefore, our study could therefore be valuable for guiding policies that promote lower readmission rates, a key element in strategies to enhance the efficiency and quality of hospital operations [4,5,6,7,8,9].

To carry out these objectives, we use a sample of Spanish public hospitals observed during 2002–2016. According to data from the Ministry of Health (2023), the percentage of readmissions in public hospitals in the 30 days following discharge was 6.9% in 2002; 7.4% in 2012 and 8.3% in 2019, which shows the magnitude of the problem in this country and its increasing evolution over time.

Methods

The Theoretical Model

In order to estimate the marginal cost of readmissions, we require knowledge of the hospital technology. Several possibilities are available for this aim. Given its multi-product nature, the most standard in the literature is the use of the cost functions [10,11,12,13,14]. In this paper, we propose the use of an input-oriented distance (D_I) function [15, 16] given that it offers significant advantages over the production and cost functions.¹ Although some studies have applied the distance function to capture hospital technology [17,18,19], these have been more focused on the analysis of hospital efficiency rather than on a study of production costs.

To explain the model, we define the input distance function D_I that describes the maximum possible equi-proportional reduction of a vector of inputs necessary to produce a vector of given outputs. Following [20], the input-oriented distance function fulfils the following properties: it is non-decreasing in inputs; decreasing in outputs; homogeneous of degree 1 and concave in inputs. Furthermore, the model makes explicit use of the variable R that indicates the rate of unscheduled hospital readmissions (less than 8 days after discharge). It is worth mentioning that although public hospitals cannot control the number of patients admitted, the latter are conceivably treated with a greater or lesser degree of care depending on the different medical or management practices followed by the hospital. Therefore, in this study readmissions are considered as potentially endogenous which may result in biased results.

To summarise, the proposed model starts from an input-oriented distance function that can be defined as a function of a vector of inputs (x), outputs (y), and the variable R defined above. Formally:

$$\text {D}_\text{I}(\text{x,yR})=\text{max}_{\delta}\{\delta:(\text{x}/{\delta})\geq\text{y}\}\delta\geq1$$

(1)

In Eq. (1), $\updelta ={D}_{I}$ represents the maximum reduction in the vector of inputs (x) that allows the vector of outputs (y) to continue to be produced. When D_I is equal to unity, production is technically efficient. A value greater than one indicates the degree of (in)efficiency achieved.

Given that D_I is homogeneous of degree 1 in inputs, it is possible to arbitrarily choose one of them (e.g. x₁) to impose this property. Then, and considering D_I in logarithms we obtain:

$$-\text{ln}{x}_{1}=ln{D}_{I} \left(\frac{x}{{x}_{1}},y,R\right)- ln\updelta$$

(2)

Renaming lambda: $u=ln\updelta$, we get:

$$-\text{ln}{x}_{1}=\text{ln}{D}_{I} \left(\frac{x}{{x}_{1}},y,R\right)-u$$

(3)

In the following, we explain how the duality between the input-oriented distance function and the cost function allows us to obtain both, the readmission marginal cost and the output marginal costs. To do this, first, the prices of the factors of production in the cost function are normalised so that their value will now be equal to unity. In other words, $C\left(W,y,R\right)=1;$ where $\text{W}=\frac{w}{C\left(w,y,R\right)},$ and w is the input price vector. Following [21], it is possible to establish a relationship between the input-oriented distance function and the cost function as follows:

$${D}_{I}\left(x,y,R\right)=\underset{W}{\text{min}}\left\{Wx:C\left(W,y,R\right)=1\right\}$$

(4)

$$C\left(W,y,R\right)=\underset{x}{\text{min}}\left\{Wx:{D}_{I}\left(x,y,R\right)=1\right\}$$

(5)

Equations (4)- (5) define the Shephard duality between the two functions. Applying this duality, we can obtain (see Annex I for details):

$$\frac{\partial C\left(w,y,R\right)}{\partial R} =-\frac{\partial {lnD}_{I}\left(x,y,R\right)}{\partial lnR}\frac{C}{R}$$

(6)

Equation (6) quantifies the marginal cost of readmissions. The sign of the derivative in (6) will indicate its effect on hospital technology (on the isoquant). Thus, a positive sign indicates additional costs, meaning that readmissions are associated with a higher cost of care (shifting the isoquant to the right), and vice versa.

Similarly, the marginal cost for the output y_r is:

$$\frac{\partial C\left(w,y,R\right)}{\partial {y}_{r}}=-\frac{\partial {lnD}_{I}\left(x,y,R\right)}{\partial ln{y}_{r}}\frac{C}{{y}_{r}}$$

(7)

In addition, following [22], the proposed model allows us to obtain technical efficiency (TE) indices with which to compare the activity of different hospitals as follows:

$$TE= \mathit{exp}\left(-u\right)$$

(8)

In other words, TE represents the maximum possible radial reduction in input consumption, in order to continue obtaining a given level of output. The TE index takes values between 0 and 1, with values close to one indicating higher levels of efficiency. An index value equal to 1 would indicate that the hospital is technically efficient.

The empirical model

We propose the estimation of Eq. (3) via the use of a stochastic frontier model [23]. Using a panel of Spanish NHS hospitals observed over the period 2002–2016, and assuming a log-linear Translog functional form, Eq. (3) can be rewritten as:

$$\begin{array}{c}-\text{ln}{x}_{1ht}={\alpha }_{0}+\sum_{r=1}^{m}{\alpha }_{y}\text{ln}{y}_{i}+\frac{1}{2}\sum_{r=1}^{m}\sum_{s=1}^{m}{\alpha }_{rs}\text{ln}{y}_{rht}\text{ln}{y}_{sht}+\sum_{i=1}^{n-1}{\beta }_{i}\text{ln}\frac{{x}_{iht}}{{x}_{1ht}}+\frac{1}{2}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}{\beta }_{ij}\text{ln}\frac{{x}_{iht}}{{x}_{1ht}}\text{ln}\frac{{x}_{jht}}{{x}_{1ht}}+\sum_{r=1}^{m}\sum_{i=1}^{n-1}{\rho }_{ri}\text{ln}{y}_{rht}\text{ln}\frac{{x}_{iht}}{{x}_{1ht}}\\+{\pi }_{R}\text{ln}{R}_{ht}+{\pi }_{RR}{\frac{1}{2}(\text{ln}{R}_{ht})}^{2}+\sum_{i=1}^{n-1}{{\pi }_{Ri}\text{ln}R}_{ht}\text{ln}\frac{{x}_{iht}}{{x}_{1ht}}+\sum_{r=1}^{m}{{\pi }_{Rr}\text{ln}R}_{ht}\text{ln}{y}_{rht}\\+\sum_{t=1}^{T-1}{\gamma }_{t}{D}_{t}+\sum_{h=1}^{H-1}{\gamma }_{h}{D}_{h}+{\gamma }_{V}V-{\text{u}}_{ht}+{\text{v}}_{ht}\end{array}$$

(9)

where: h is the number of hospitals; t is time (in years); y(y_r, y_s) = vector of outputs (r, s = 1,…, m); x(x_i, x_j) = vector of inputs (i, j = 1,…, n); R represents readmissions; α₀ = constant; D_t are time dummies; and D_h hospital dummies. In addition to the inputs and outputs, other variables that can influence healthcare technology (vector V) were added. Specifically, the following variables have been considered: case-mix; whether the hospital is a teaching hospital; and the size of the hospital, measured by the number of operational beds.

Regarding the stochastic part of the model, standard in the frontier literature, we assume that ${v}_{ht}$ follows a zero-mean normal distribution with constant variance ${v}_{ht}\approx iid N\left(0,{\upsigma }_{v}^{2}\right)$ that captures random shocks and measurement errors. The non-negative term ${u}_{ht}$ is assumed to follow a (heteroscedastic) half-normal distribution, i.e. ${u}_{ht}\approx iid {N}^{+}\left(0,{\upsigma }_{u}^{2}\right)$. That is, u is a zero-mean normal distribution with non-constant variance truncated from below at zero. According to [24], ignoring heteroscedasticity in the error term u_ht will lead to biased estimates since homoscedastic frontier models might seriously bias both the frontier parameters and the efficiency indices. Because of this, we follow [25] and [26] by assuming that the variance of the pre-truncated normal variable depends on a set of covariates that could explain the differences between observed and potential input quantities. That is to say:

$${\sigma }_{uht}^{2}=exp({\Gamma}_{ht,}\phi)$$

(10)

where we model the variance of ${u}_{ht}$ as a linear function of a set of covariates Γ that can explain the differences between potential and observed input quantities, with ϕ being the set of parameters to be estimated. Assuming that time and the economic cycle may imply heteroscedasticity in ${u}_{ht}$, vector Γ will be a function of the business cycle measured by GDP lagged one period (which affects hospital budgets) and a trend variable.

Note that we are modelling the variance (not the mean) of the pre-truncated normal distribution in this specification. However, it should be pointed out that what we are finally modelling after the truncation is the mean (and the variance) of u. This implies that the larger ${\upsigma }_{u}^{2}$, the greater is the average distance from the frontier (see Caudill et al., 1995).

Endogeneity issues

The model is estimated via maximum likelihood. The consistency of the estimator depends on the exogeneity of the explanatory variables. However, as already mentioned, the readmissions variable is potentially endogenous. The literature on the treatment of endogeneity in stochastic frontier models is relatively recent and the solution is complicated given the special definition of the residual as a composite error term where one component is asymmetric. A possible solution suggested by [27] for the specific case of a translog functional form is to use a two-stage procedure, similar to that used in linear models where the random term is not compound. This procedure stems from Eq. (9) where it is assumed that cov (R_ht, v_ht) ≠ 0.

As a first step, R is regressed using some Z_ht instruments that are considered exogenous in the sense that E(v_ht|Z_ht) = 0. It is now possible to express the endogenous variable in its reduced form as:

$$\text{R}=\text{Z}\Pi+\eta$$

(11a)

where Z is the instrument vector and η is uncorrelated with Z. Thus the endogeneity of R implies that cov (η,v) ≠ 0. By estimating (11a) via OLS we obtain:

$$\widehat{\upeta }=R-\text{Z}\Pi$$

(11b)

The second step is to insert $\widehat{\upeta }$ into Eq. (9) and test its significance (see [28] for details).² The difference compared to a standard model is in the second step. [27, 29] show that such a procedure can also be applied for stochastic frontier models.

Other useful characteristics of technology

After estimating the model (9–10) and exploiting the duality between the input-oriented distance function and the cost function, it is possible to define the average cost elasticity (${\varepsilon }_{\overline{C },y}$) as follows [20, 30]:

$${\varepsilon }_{\overline{C },y}= \left[-\sum_{y}\frac{\partial \text{lnD}\left(\text{x},\text{y},\text{R}\right)}{\partial lny}\right]-1$$

(12)

Furthermore, by estimating the distance function it is possible to analyse how technology changes over time. Since the passage of time has been approximated by time dummies in Eq. (13), the technical change (TC) is defined as follows:

$$\text{TC}_\text{t+1}\;_\text{t}=\gamma_\text{t+1}-\gamma_\text{t}$$

(13)

where γ represents the estimated coefficients for the time dummies in Eq. (9). A positive value of TC would indicate that fewer inputs are needed to produce the output and vice versa [31].³

Data

The data comes from three databases managed by the NHS: the “Specialized Care Information System” (Sistema de Información de Atención Especializada, SIAE) which, as of 2010, replaces the “Inpatient Healthcare Establishments Statistics” (Estadística de Establecimientos Sanitarios en Régimen de Internado, ESCRI), and from the Minimum Basic Data Set (Conjunto Mínimo Básico de Datos, CMBD).⁴ These three databases have open access on the NHS website; the data are annual and cover the entire scope of public hospital activity at the national level.⁵ In order to homogenize the sample, only general public hospitals have been included in the analysis, i.e., those that can be considered to carry out a relatively similar activity and have a similar consumption of medicines and high-tech capital equipment [32].⁶ As a result of the foregoing, the final sample is formed by an unbalanced data panel consisting of 244 Spanish public hospitals observed in the period 2002–2016 (with a total of 3391 observations).

According to the theoretical and empirical models previously proposed, the estimation of the model (Eqs. 9–10) requires data related to outputs, inputs, and other control variables (see for example, [24]). Following previous studies in the literature, among others [33] or [34], we consider the following variables to capture hospital activity:

• a) Hospital activity

Output is approximated by the number of cases treated. Given that the discharge variable provided by the NHS data is highly disaggregated, it has been grouped into three categories in order to simplify the number of variables used. To achieve homogeneous groups in terms of resource consumption and complexity, we have weighted the discharges using the WCUs (Weighted Care Units) division criterion defined by the Spanish Ministry of Health (2021) and [35], as explained in Annex II. The output vector is defined as follows:

• Medical discharges (y₁): general medicine, psychiatry, tuberculosis, long stay, rehabilitation, general surgery, paediatrics and gynaecology, obstetrics, major ambulatory surgery, and others.

• Intensive care discharges (y₂): intensive care unit, burn unit, and neonatal intensive care unit.

• Outpatient services (y₃): outpatient consultations (first and subsequent) and emergencies not requiring admission.

Furthermore, as we explained previously, it is necessary to consider these hospital discharges as being heterogeneous, given that some patients may have been discharged before their complete recovery. Therefore, we introduce the rate of unscheduled hospital readmissions less than 8 days after discharge (R) in order to capture this heterogeneity.⁷

• b) Input factors

For the estimation of the model, it is also necessary to have information on the factors of production. Those considered for obtaining the hospital output are labour (divided into categories of personnel), supplies, and capital.

• Labour: The NHS data provides information on personnel broken down into the categories of medical staff (doctors, pharmacists, and other graduates), non-medical health staff (nurses, midwives, physiotherapists, and other graduates, health technicians, clinical assistants, and other care assistants), and non-health staff (management and administration personnel, skilled trades personnel, unskilled trades personnel, and other personnel). However, although initially an attempt was made to make the best use of disaggregated data, a problem has arisen in that the classification of some categories is very erratic. For this reason, the personal input is broken down into three categories: medical staff (x₁), non-medical health staff (x₂), and non-health staff (x₃). When the worker is hired part-time, it is weighted by 0.5.

• Supplies: The available data does not contain information about the number of supplies consumed by physical units, but instead obtains data via the expenditures incurred on the different consumables. For this reason, supplies are measured through expenditure in monetary units for the different concepts. In this analysis, these items have been divided into two relevant categories: pharmaceutical expenditure (x₄) and other expenditure (x₅). It is important to note, that both variables are "contaminated" by the effect of price variation. In order to resolve this problem, the monetary value of both variables has been deflated using the price index for pharmaceutical products in the first case, and the price index for sanitary products in the second (both indexes have been obtained from the National Statistics Institute).

• Capital: Two variables were used to approximate the hospital capital endowment. Firstly, the hospital operating beds (x₆) are considered as a proxy variable for the capital input. Its use is very common in this type of literature because the data on capital or depreciation are unreliable, if not null. Secondly, the number of hospital diagnostic and therapeutic tests (x₇) reported in the NHS data have been used. In order to be able to add these up, they have been weighted in line with the average price that has been agreed by the NHS with different insurance entities for each test.⁸

• c) Other variables

The database used incorporates a diverse range of hospital characteristics. It includes small and large hospitals, isolated county hospitals, and others located in large cities. It is evident that the activity of hospitals, with such distinct characteristics, may contain differential elements that condition management efficiency and that are not directly attributable to management. Furthermore, the criterion used to measure hospital output, the number of cases treated, does not consider the different complexity of the patients treated. However, in Spain there are more complex hospitals, with more technology, which treat more complicated cases.

The use of a panel of data partially solves this problem by capturing the specific effects of each hospital that remain constant over time. However, these effects do not contemplate the variations in complexity experienced by the hospitals during the period considered. For this reason, different variables have been introduced with the aim of capturing this complexity. Firstly, the STUDENTS variable is expressed as the number of medical students who work in the hospital. In the literature on the hospital sector, it has been common to introduce this variable as a proxy for complexity, since teaching hospitals are generally the largest and most complex, are located in metropolitan areas and have a high investment in technology (see for example [36]). Secondly, the CASE-MIX variable, built by the NHS, has been included. The variable indicates the complexity of cases through the average cost of discharge per patient. To do this, patients are grouped by DRGs (Diagnosis Related Groups—each group has an associated cost), and the average cost per DRG is expressed weighted by the number of patients relative to the overall average cost. This variable takes values between zero and one.⁹ A value close to 1 indicates that the complexity of the cases treated by the hospital is high and, therefore, the costs, are higher. Finally, a dummy variable has been included to control for technological differences in hospitals with less than 100 beds (H < 100beds).

Additionally, in Eq. (9) a vector of binary variables has been added for each year considered. This vector represents variables that, evolving over time, affect all hospitals in the same way, and can be used, for example, to evaluate the effect of two regulations that have affected hospitals during the period under analysis, specifically, their pharmaceutical expenditure: the first, is the Royal Decree 8/2010 published by the Ministry of Health, Consumption and Social Welfare concerning the list of drugs to which a deduction in price is applicable; the second refers to the irruption in 2014 of the treatment for Hepatitis C and the subsequent implementation of the national strategic plan to address it.¹⁰

Regarding Eq. (10), in order to analyse the effect of time on TE, GDP was lagged one period to capture the impact of the cycle (GDP_1) together with a time trend. Finally, in order to calculate the marginal costs, the variable Costs (C) is obtaining as the sum of hospital expenses and purchases (purchases and supplies; personnel costs; external services). By way of summary, Table 1 presents a brief definition of the variables used in the estimation. Table 2 shows the descriptive statistics.

Table 1 Definition of the Variables

Table 2 Descriptive Statistics

From Table 2, it is observed that, on average, the percentage of readmissions less than 8 days after discharge is nearly 3%, reaching a maximum of 11.43%. The average number of students in university hospitals is around 70, although in some hospitals, this number exceeds 500 students. Only 18% of the sample are hospitals with fewer than 100 beds, with the average hospital size having around 354 beds. Pharmaceutical expenses represent 14.2% of variable costs, while expenses on diagnostic tests represent 9.5%. The case-mix index which takes a value of 0.51 at the sample mean, indicates that the average complexity is moderate (values close to 1 indicate moderate complexity, while values above 1 indicate high complexity).

Results

The model (9–10) has been estimated via Maximum Likelihood, using the econometric program Stata 16. All continuous variables have been taken in deviations with respect to their means (approximate functional form), so that the first-order parameters presented in Table 3 reflect the elasticities of the distance function at the sample mean.

Table 3 Distance Function Estimated

The coefficients related to inputs and outputs are significant and have the expected sign, so that the estimated distance function fulfils the theoretical properties required of being decreasing in outputs and non-decreasing in inputs. The residual calculated according to Eq. (11b) has been included at a second stage in Eq. (9). As instruments (vector Z), we use the endogenous variable lagged one period (predetermined variable) and the exogenous variables in Eq. (9), including the time and hospital dummies.¹¹ The coefficient of this residual is significant, confirming the endogeneity of the R variable and the need to carry out the adjustment.

As far as complexity is concerned, teaching in hospitals, measured by the number of students, presents a negative and significant elasticity. The negative sign of the estimated coefficient implies that university hospitals, ceteris paribus, need more factors to carry out their activity. These types of hospital, in addition to having the additional activity of training younger students [36], are generally the largest and most complex, maintaining a high investment in technology and attending the most serious patients. Moreover, a size dummy has also been included for hospitals with less than 100 beds, in order to control for differences in the care activity of these hospitals. The estimated coefficient is positive and significant, indicating that these hospitals need 9.2% less productive factors than larger hospitals in order to obtain a given number of discharges. Finally, the coefficient of the CASE-MIX variable was not significant at the frontier.¹²

Regarding the effect of readmissions on healthcare technology and costs, the coefficient of the variable R shows a negative and significant sign in Table 3. This implies that readmissions shift the isoquant to the right, meaning that hospitals with readmissions have higher marginal costs than hospitals without readmissions, ceteris paribus. Its significance confirms the specification of the model that considers R as a relevant factor for the hospital, the omission of which would be biasing the results (see Annex III for more details).

In sum, this result indicates that it is more costly for a hospital to discharge a patient prematurely (with the consequent readmission) compared to keeping the patient in hospital until a definitive cure is achieved. It seems reasonable: the last days of any hospital stay are usually the least demanding in terms of resources and therefore the least costly.

Marginal Cost Analysis

As seen in the theoretical model, the application of the duality between the input-oriented distance function and the cost function implies that it is possible to recover relevant characteristics of technology such as marginal costs. In the sample mean and according to Eq. (7), the marginal costs of the outputs are: 715.3 € for medical discharges (y₁); 32,437.16 € for ICU discharges (y₂) and 238.85 € for outpatient consultations (y₃). These results are in line with those obtained by [33] where a cost for medical discharge of 768.89 €, and 92.28 € for outpatient consultations was obtained.¹³ Figure 1 shows the evolution of these costs for the three outputs analysed. In all three cases, an increasing trend seems to be observed, especially in periods of expansion, where healthcare may be associated with greater use of inputs. Also, the case of outpatient consultations is relevant, where marginal costs skyrocket after the introduction of the hepatitis C treatment.

Fig. 1

Evolution of the Marginal Cost of hospital discharges and consultations

The main objective of our study is to shed some more light on the effect of readmissions on hospital technology and costs, by estimating the marginal cost of the readmissions. From Eq. (6) it can be deduced that, on average, an increase in unscheduled readmission implies an increase in variable costs of 1,334 €. This cost has a generally increasing trend, as shown in Fig. 2, and, as in the case of hospital outputs, a greater increase is observed in periods of expansion.

Fig. 2

Evolution of the Marginal Cost of Readmissions

It is also interesting to analyse this cost whilst bearing in mind the size of the hospital. Thus, Table 4 shows that the largest hospitals are the ones associated with higher costs of readmissions, given that they attend the most complex cases and, therefore, require greater use of resources. Thus, while the cost of a readmission in small hospitals is around 260 €, this cost soars in larger hospitals with more complex cases, reaching more than 4,000 € for those with more than 1,000 beds.

Table 4 Marginal Cost of Readmissions by Hospital Size

There are also relevant differences if we also consider the teaching activity of the hospital. Thus, Table 5 shows a marginal cost of an additional readmission of 383.2 € in a non-university hospital, compared to 1,703.5 € in a teaching hospital.

Table 5 Marginal Cost of Readmissions: Teaching hospital vs non-teaching hospital

Technical Efficiency Analysis

Table 6 presents the results relating to the determinants of the variance for the error term $u,$ following Eq. (10).

Table 6 Estimated error terms and TE Determinants

A negative sign is interpreted as a reduction in the distance that inefficient hospitals are from the frontier and vice versa. Results indicate that the technical efficiency of hospitals significatively improves over the period. In addition, similar to that which occurred when analysing the marginal costs of hospitals located on the technological frontier, the inefficiency of hospitals located within the frontier increases in periods of expansion, where healthcare may be associated with a greater availability of resources.

As defined by Eq. (8) it is possible to calculate the TE indices. It is worthwhile remembering that the TE indices take values between 0 and 1, with a value equal to 1 indicating that the hospital has reached its production frontier. The average TE for hospitals is 0.97, which would imply that, on average, 3% less productive factors could have been used to still produce the maximum potential output. Figure 3 shows the evolution of the average efficiency of the hospitals for each year, this displaying an increasing trend since the beginning of the period.

Fig. 3

Temporal evolution of technical efficiency

Other characteristics of the technology

From the coefficients presented in Table 3 and Eq. (12), we obtain a value of average cost elasticity equal to 0.67 and significantly lower than unity, which would confirm that hospital services production technology in Spain presents economies of scale.¹⁴ That is, average costs decrease as production increases, which would imply a productive advantage of large hospitals over smaller hospitals, due, for example, to a greater division and specialization of factors.

Regarding technical change, from Eq. (13) we obtain the TC rates presented in Table 7.

Table 7 Technical Change (TC)

For most years, the indices have a negative sign, indicating that the passage of time implies the need for a greater use of productive factors. However, this does not mean that time negatively affects hospital technical progress. On the contrary, it is technological progress that may explain this result. Indeed, the rapid pace at which scientific advances are developing is allowing innovation in new drugs and technologies, and this is becoming more pronounced as we move forward in time. However, the cost of researching and developing each new technology or drugs is very high. For this reason, and due to the fact that in the estimated model pharmaceutical products and diagnostic tests are expressed in monetary terms, the result obtained may be reflecting the effect of the high price of these cutting-edge technologies and drugs. Thus, for example, the high coefficient observed in Table 7 referring to the last two years of the study may be explained by the implementation of the Hepatitis C treatment from 2014 onwards.

Discussion

As [37] points out, overseeing healthcare spending can lead to savings without necessarily impacting on patient health. Therefore, studying how hospital activity affects production costs and the potential for savings on unnecessary expenses seems relevant. This study introduces a novel model in the literature that enables the analysis of the impact of hospital readmissions on costs. Consequently, since our findings suggest that readmissions incur additional costs for the hospital system, they need to be minimized, thereby justifying the design and implementation of specific policies related to readmission rates.

Thus, our study could be pertinent for shaping policies aimed at incentivizing their reduction. Indeed, readmission rates constitute a significant aspect of ongoing strategies to enhance hospital sector activity. Consequently, there are examples of policies being implemented to reduce readmission rates with positive outcomes. Examples include, [7] for Costa Rica, [38] for Israel, or [4] for USA who found that these policies not only decreased readmission rates¹⁵ but also resulted in an improvement in hospital quality. On the other hand, [5] points out that variations in reported levels of readmission are likely related to variations in the unobservable quality of hospital services. At the European level, initiatives aimed at reducing readmissions have also emerged such as those implemented in Denmark in 2002 and 2006, in the UK in 2001 and 2011, or in Germany in 2004 [3].

In addition to financial incentives (for example, including bonuses for "good performance," such as reducing the number of rate of readmissions, or penalizing hospitals with higher readmission numbers or rates), [44] highlight other measures that can be implemented to help reduce avoidable readmissions. These initiatives include investment in quality, utilizing available technology for efficient patient record-keeping, or telemonitoring patients, among others. Regarding the specific case of Spain, while concrete programs aimed at reducing readmission rates have not been implemented so far, the information gathered in this study may prove highly relevant for future regulatory policies.

In any case, our results can assist hospitals in managing these rates effectively, reducing the need for unnecessary and costly readmissions, thus enhancing the efficiency of the hospital care system. For example, to keep patients until they can be discharged from the hospital with certain guarantees; and a follow-up of them by the hospitals or by the primary care service could prove useful. For example, the NHS can identify and target patients who are at higher risk of readmission. This includes elderly individuals, patients with complex medical and social needs, and those with limited cultural and economic resources that pose a risk to maintaining their health. This strategy would involve close monitoring and support, ensuring that these patients receive the necessary care and resources to manage their health effectively after discharge, ultimately improving their outcomes and reducing the strain on hospital services. Implementing and analysing the results of these strategies would undoubtedly be a step forward in the search for solutions that serve to reduce readmissions that, as shown in this study, represent a significant cost for the national health system.

Finally, it is worth nothing that in this empirical example for Spain, our sample includes Spanish public hospitals with the specialties of general, oncological, medical-surgical, surgical, maternal, paediatric, and maternal-child. Thus, the results of this study can be useful for this type of public hospital, but caution is necessary when extending these results to private hospitals or those with other specialties. Additionally, the representativeness of our results may be limited in countries with regulations regarding readmissions that differ from the Spanish case and with different health system models.

Conclusions

The objective of our research was to analyse, both theoretically and empirically, the extent to which hospital unscheduled readmissions (within 8 days of discharge) can impact hospital technology and costs. This study proposes a model that considers the existence of two types of readmissions: those involving patients who have been discharged after complete recovery and those who left the hospital before completing recovery and are subsequently readmitted. These different discharge scenarios require different inputs and, therefore, entail different costs. In this paper, we have addressed the question of which type of discharge is more costly for the hospital, considering that readmissions can be endogenously determined by hospitals depending on how they care for patients. To achieve these aims, we have utilized the existing dual theory between cost and distance functions to estimate the marginal cost of readmissions, which, to our knowledge, represents a novelty in the literature.

Using the panel data of Spanish public hospitals, the results obtained allow us to conclude that the treatment required by a patient who is readmitted is more expensive than keeping the patient in hospital until achieving a definitive recovery and discharge. The latter makes sense if we take into account that the last days of a hospital stay are the least demanding in resources and therefore, the least expensive. Moreover, we find that this additional cost follows an increasing temporal trend, especially in times of expansion when the use of resources is greater.

This kind of exercise can also help policymakers at regional level in Spain to analyse how to establish Key Performance Indicators (KPIs) for monitoring the management agreements signed between public hospitals and regional administrations. Among the most frequent KPIs included in these contracts and those that most concern hospital managers are readmission and discharge rates. But these indicators are not particularly sophisticated and perhaps should be more detailed. For example, an interesting indicator could be readmission rates for early discharges.

It can also assist in the design of new hospitals and care processes. Currently in public hospitals the bed occupancy rate is very high and there are hardly any beds available, so discharging patients is one of the main concerns. Additionally, it can help managers quantify the inefficiency and costs associated with early patient discharges, allowing them to compare these with those of other areas in order to prioritize management strategies.