List size disequilibria and service provision in general practice

: Several Nordic countries remunerate general practice by a mix of capitation and fee-for-service. From the literature we know that capitation-based payments come with a risk of undersupply of services, whereas fee-for-service comes with a risk of overprovision of services. Previous studies from the Nordic countries assess potential overprovision of services in general practices that are falling short of enlisted patients. However, today the main challenge in general practice is physician shortages, which comes with a risk of underprovision of services. Little is known about whether physician shortage in fact leads to underprovision of services. Using the two-way Mundlak regression on a panel of Danish general practices in 2016-17, this study assesses whether holding a longer than preferred patient list is associated with fewer services per enlisted patient. Around 100 of our sample of 1,652 practices hold longer lists than preferred. These practices have on average an excess of around 80 patients per full time general practitioner. We find little support of the hypothesis that practices with longer than preferred lists provide fewer contacts per patient. Heterogeneity analyses, however, reveal that practices with longer lists tend to provide fewer services to patients with complex needs. Policymakers should therefore be aware that there may be underprovision of services to high-need patients when there is a shortage of GPs.


Introduction
Economists model physicians' responses to different remuneration schemes by assuming that they maximize their utility while facing a trade-off between income, patient benefits, and leisure (Léger, 2008, McGuire, 2000, Scott, 2000).Several Nordic countries remunerate general practice based on a mixed scheme (Olsen et al., 2016).Mixed remuneration schemes are shown to be superior to pure fee-for-service (FFS) and pure capitation-based schemes as they reduce the risk of both under-and overprovision (Ellis and McGuire, 1986).Underprovision in capitation-based payment schemes has been shown theoretically and in laboratory experiments to be skewed towards patients with higher needs for care (Barham and Milliken, 2015, Brosig-Koch et al., 2017, Hennig-Schmidt et al., 2011, Oxholm et al., 2019).However, little is known about this underprovision in a real-world setting and in mixed payment models.As general practitioner (GP) shortage is increasing in many countries (among those the Nordic countries) (Oxholm et al., 2021, Aas et al., 2021), it is important to assess the risk of underprovision in mixed remuneration schemes.
The basic economic model assumes that physicians are free to choose a list size and level of service to patients.According to this model physicians reach an equilibrium choice where their marginal utility gain equals their marginal utility costs of providing care (Scott, 2000).In real life, however, some regulatory or market constraints may prevent this equilibrium level of care.Obvious constraints are patients' lack of demand for services, competition for patients, regulatory rules for minimum and maximum list sizes as well as shortage of physicians.A result of these constraints are GPs holding either shorter or longer lists than preferred, resulting in a disequilibrium.Iversen (2004) studies the effect of such constraints by analysing both theoretically and empirically the effect on service provision of GPs holding a lower than preferred list size in a mixed remuneration scheme.Iversen (2004) finds that GPs with patient shortage generate more fee-for-services per patient, thereby contributing to the literature of income-motivated behaviour, which could be a sign of overprovision of care (Grytten et al., 1995, McGuire, 2000, Di Guida et al., 2019).
As opposed to the beginning of the millennium, many countries, including the Nordic countries, are today challenged by a shortage of GPs rather than shortage of patients (Olsen et al., 2016).This challenge is further intensified by the demographic developments, where people live longer but with more chronic diseases (Willadsen et al., 2016).Policy concerns have therefore changed from fear of overprovision towards fear of underprovision.Consequently, policymakers are discussing if the mixed payment scheme can be adjusted to meet these new challenges.Both Sweden and Denmark now differentiate capitation payments to take different (and increasing) workloads in general practices into account (Anell et al., 2018, Danish Regions' Committee for Salaries and Fees (RLTN), 2021), and this adjustment is currently being discussed in Norway as well (Melbye, 2022).
This study aims to assess whether general practices holding a longer list than preferred are providing fewer services per patient after controlling for patient characteristics.We use a panel of almost all Danish practices from 2016 and 2017 (N=1,652) with vast information on enlisted patient characteristics (socio demographics and morbidity), actual list size, preferred minimum and maximum list size, level of service provision, and geographical location.

Institutional setting
Danish GPs are self-employed working on a contract negotiated in a collective agreement with the public healthcare system (the Regions).The collective agreement is (re)negotiated every third year.The agreement details not only the reimbursement, but also the opening hours and rules about the number of patients they (as a minimum) should be willing to assign to their clinic (Pedersen et al., 2012).The general practices are reimbursed by a mix of FFS and capitation-based payments, where the FSS part constitutes most of the total payment received by the practice (in 2016 approximately 70%) (Olsen et al. 2016).During our analysis period (2016)(2017) the size of the capitation payment was the same for all patients, regardless of their characteristics.
All citizens should be registered with a general practice.On average practices have enlisted approximately 1,600 patients per GP.This number rests on the regulatory rule that the list size is held open if it is less than 1,600 patients per full time GP, unless exempted by the authorities.At the same time, practices need to apply for exemptions if they want to hold a list size above 2,700 patients per GP.Despite these rules, the list size varies quite substantially among practices (in 2017 mean 1,681; standard deviation 324).In 2017 around 60% of the practices operate with a closed list.Practices can report the minimum and maximum list size they want to hold before they would prefer the list respectively opens and closed.However, these reported preferences do not necessarily imply that the list is about to open or close.Therefore, some practices' actual list size may be shorter or longer than their reported preferences.Around two out of three practices have reported their preferred maximum list size.In 2016 around 10% of the practices changed their preferred maximum between January and December whereas in 2017 around 20% changed their maximum.
On average there are two GPs plus nurses and secretaries in a practice (The Danish Organization of General Practitioners, 2021).Approximately, half of the practices operate as solo practices.The GP acts as a gatekeeper to more specialised healthcare and also manages most monitoring of patients' chronic diseases.The GP handles the majority of consultations without further referral (Pedersen et al., 2012).Health care in Denmark is taxfunded and GP and hospital services are free of charge.The healthcare system is embedded in a decentralized administrative structure of five regions (Olsen et al., 2016).As is the case in many countries, there is a rising concern of GP shortages in Denmark with the number of GPs slightly falling over the last decade coinciding with a rise in the size of the Danish population (Oxholm et al., 2021, The Danish Organization of General Practitioners, 2022, The Danish Organization of General Practitioners, 2023).An increasing number of GPs are also reaching retirement age (O'Halloran et al., 2021) and it is difficult to attract new GPs in certain rural areas.General practitioners average age is 51.5 years and approximately 9% are over 65 years old, but this share varies from 0% to 50% at municipality level (The Danish Organization of General Practitioners, 2021).

Data
We use a nationwide cohort of general practices operating in both 2016 and 2017 (N=1,652) identified in the Provider register.We exclude general practices going from or to being a solo practice (N=18).We keep practices who go from having 2-3 GPs to having at least four GPs.However, we exclude these practices in subgroup analyses (N=24).From the Provider register we have information on practice organisation (solo or group practice), the number of GPs in the practice, the geographical location of the practice, the practice's total list size, the practice's preferred minimum list size, their preferred maximum list size, and whether they have closed their list for new patients.By merging on data from the National Health Service Register (Andersen et al., 2011), we obtain information on which patients are enlisted with each practice, which services the practices' provide to their patients, and the total revenue per full time GP in a practice.To characterise practices' enlisted patients we merge on a number of registers from Statistics Denmark (Register on education (Jensen and Rasmussen, 2011), Income Register (Baadsgaard and Quitzau, 2011) and National Patient Register (Lynge et al., 2011)).From these registers we have detailed measures of patients' socioeconomic statuses (measured as the share of patients on the list with the lowest level of education) and health statuses (measured as the share of patients with a score of more than one in the Charlson Comorbidity Index (Quan et al., 2011) as well the share of patients with more than one of the most common chronic diseases1 (The Danish Health Data Authority, 2022)).

Methods
We follow the approach used in Iversen (2004) using panel data models to assess the effects of deviations from the preferred list size on general practices' provision of services per enlisted patient.As mentioned in the introduction, Iversen (2004) develops a theoretical model of physician behaviour in mixed remuneration schemes, which assumes the ability to reach the preferred list size is a key determinant of the level of service provision per patient (Iversen, 2004).More specifically, Iversen (2004) hypothesizes that rationed GPs, i.e., GPs facing a shortage of patients, are likely to provide beyond the minimum acceptable intensity of services.Iversen (2004) confirms this hypothesis empirically using a five-year panel of GPs, which includes GPs' preferred list size.Iversen define patient shortage by GPs having a shorter list size than their reported preference.Iversen analyses the consequences of this patient shortage by applying both random effect models and difference-in-differences models.
We deviate from Iversen (2004) by studying the consequences of GP shortage instead of patient shortage.We define our treatment variable 'GP shortage' as holding a longer list than the reported preferred maximum list size.Associations between GP shortage and service provision may arise from variation over time when a practice experience a change in GP shortage (within-practice variation) as well as from variation across practices' experiencing different degrees of GP shortage (between-practice variation).With only a two-year panel, we expect the within-practice variation to be limited.We therefore make use of the Mundlak regression model, which estimates both the within-and betweenpractice variation in service provision (our dependent variable) due to GP shortage (our treatment variable) (Mundlak, 1978): Where vjt is the number of contact services per patient for practice j at t=2016, 2017.   is a dummy that equals one if the list size exceeds the preferred maximum in year t, and    ̅̅̅̅ is the average of this dummy over years .The estimate  1 is the within treatment effect while the estimate  2 is the between treatment effect.  is a vector of patient characteristics associated with their need for services at year t (e.g.share of patients on the list with chronic disease, the Charlson Comorbidity Index, socio demographics etc.),   ̅ is a vector of average values of the patient characteristics over years t,  is a constant,   is the random intercept for practice j assumed to be normally distributed, whereas   is the idiosyncratic error term.

Heterogeneity analysis
As studies show that especially services to high-need patients is at risk of rationing (Hennig-Schmidt et al., 2011, Oxholm et al., 2019), we assess patient heterogeneity treatment effects by splitting the outcome variables (number of services per patient) in eight subgroups of patients according to their morbidity and sociodemographic profile.First, we look at a) patients with a score of more than one in the Charlson Comorbidity Index (Quan et al., 2011), and b) patients with more than one of the most common chronic diseases (se footnote 1).Next, for each of these two groups of patients we define subgroups of patients by i) being 18-59 years old and receiving transfer income, ii) having a diagnosis of alcohol or drug abuse or a diagnosis of depression or schizophrenia, and iii) both i and ii.Hence, in total we identify eight subgroups of patients for which we estimate the effect of GP shortage on the services they receive.We also assess heterogeneity in the treatment effect across various subgroups of practices (solo practices, group practices (2-3 GPs and 4+ GPs), and practices operating in areas that the health authorities define as threatened by GP shortage).

Robustness analyses
In a robustness analysis we test for a so-called 'dose-response' effect of GP shortage by assessing if service provision differs in practices with larger deviations from their preferred list size (31+ patients) as compared to practices with lower deviations (1-30 patients).This 30-patient threshold is a pragmatic choice as it splits the treatment group in two equal sized groups and as it approximates to a substantial share (approximately 2%) of the average list size per GP.In another robustness analysis, we redefine the treatment variable for 'GP shortage'.In our base analysis we define treatment as practices holding a longer than the maximum preferred list.However, the accuracy of this measure may be threatened by the practices not continuously reporting their preferences.Thus, the baseline measure of holding a longer list than preferred may not perfectly capture GP shortage.Instead, we now define 'treated practices' as holding a longer list than what their enlisted patients' characteristics predict: Where (  |  ) is estimated from the predicted values of Ljt from a linear regression of   on   .An additional argument for this alternative treatment definition is an increased statistical power as we do not depend on practices having reported their preferred list size.

Descriptive statistics
Table 1 shows descriptive statistics for the general practice population in the last year in our dataset (2017).The average practice generates a revenue of DKK 2.2 million 2 per fulltime GP, serves around 1,680 patients per full time GP, with a total list size of around 3,200 patients, 49% operate as solo practices, and 10% are located in an area that the health authorities define as threatened by GP shortage.General practices provide 6.6 contact services (face to face-, email-, telephone, and annual control consultations) per enlisted patient per year (v) of which around half of the visits are face to face consultations.Around 25% of the patient population has primary school as their highest level of education, 22% are retired, and around 6% are on transfer income.Around 86% are ethnic Danes, 3% have a Charlson Comorbidity Index score of more than one, and the average prevalence of COPD and type 2 diabetes, dementia, osteoporosis, and schizophrenia are 3%, 4%, 0.7%, 3%, and 0.6%, respectively.The prevalence of patients with more than one of these chronic conditions (chronic 2+) are 2.5% whereas 1.5% have diagnoses related to abuse or a mental challenge.
2 The exchange rate is 1 DKK = 0.13 EUR Notes: contacts refer to the average number of contact services per patient in the year in total (all) and subdivided into respectively face-to-face, email, telephone, annual control and preventive home visit, Solo clinic is the share of clinics operated as solo, List size is the actual list size, L_max is the share of practices with an actual list size larger than the preferred, L_min is the share of practices with an actual list size lower than the preferred.Area lacking GPs is the share of practices located in a geographical area that the health authorities define as threatened by GP shortage.Low education is the share of patients with primary school as highest educational level.Low income is the share of patients in the first quartile of the disponible income distribution (DKK).Retired is the share of retired patients.Ethnic Danes is the share of patients categorized as ethnic Danes.Dementia, Diabetes, COPD, Osteoporosis, and Schizophrenia are shares of patients with the given disease.Chronic 2+ is the share of patients with at least two of the following chronic diseases: asthma, dementia, COPD, RA, osteoporosis, schizophrenia, type 1 diabetes, and type 2 diabetes.Charlson 2+ is the share of patients with a Charlson index for at least 2. Transfer income is the share of patients between 19-59 that receive transfer income and abuse/mental is the share of patients that according to the Elixhauser comorbidity index have had either an alcohol abuse, a drug abuse, a psychosis, or a depression within the last 5 years.
Figure 1 shows the distribution of the variables used to construct the exposure of interest, i.e., the difference between the practices' actual list size and their preferred maximum list size.The number of practices with an actual list size above their preferred maximum is 63 in 2017 (approximately 4% of our total practice population and 6% of the practice population with a reported maximum preferred list size) and their average deviation is 87 patients per GP (approximately 5% above the sample mean).The preferred list size has an implicit constraint of 2,700 patients per GP due to the regulation mentioned in section 2. As seen in the histogram most practices have small deviations from their preferred list size and the distribution is right-skewed.In 2016 (data not shown) 54 practices actual list size exceeds the preferred maximum.The average deviation from the preferred list size is 77 patients per full time GP (approximately 5% above the sample mean).Table 2 shows how the exposed practices in 2017 differ from practices with actual list sizes below their preferred maximum list size.For most observable characteristics we see no statistically significant differences between the two groups, although practices with higher than preferred list size seems to have slightly more deprived patients as they have a larger share of patients with low education and of patients being on transfer income.Surprisingly, there is no difference in their actual list size per GP (L).Hence, practices with a longer list than they prefer are not as such holding statistically significantly longer lists.However, they are more likely to operate in areas that the health authorities define as threatened by GP shortage (22% as compared to 10%), facing an older patient population (24% retired patients as opposed to 21%), and they provide 0.3 more face to face services per patient.N 63 1,087 Notes: contacts refer to the average number of contact services per patient in the year in total (all) and subdivided into respectively face-to-face, email, telephone, annual control and preventive home visit, Solo clinic is the share of clinics operated as solo, List size is the list size per capacity, Area lacking GPs is the share of practices located in a geographical area that the health authorities define as threatened by GP shortage.Low education is the share of patients with primary school as highest educational level.Low income is the share of patients in the first quartile of the disponible income distribution.Retired is the share of retired patients.Ethnic Danes is the share of patients categorized as ethnic Danes.Dementia, Diabetes, COPD, Osteoporosis, and Schizophrenia are shares of patients with the given disease.Chronic 2+ is the share of patients with at least two of the following chronic diseases: asthma, dementia, COPD, RA, osteoporosis, schizophrenia, type 1 diabetes, and type 2 diabetes.Charlson 2+ is the share of patients with a Charlson index for at least 2. 502 practices have missing L_max and are therefore not included.Transfer income is the share of patients between 19-59 that receive transfer income and abuse/mental is the share of patients that according to the Elixhauser comorbidity index have had either an alcohol abuse, a drug abuse, a psychosis, or a depression within the last 5 years.

Regression analyses
Table 3 shows the results of the regression analyses.We apply a stepwise inclusion of the control variables to assess their influence on the within and between estimates of the association between service provision and having more enlisted patients than preferred.Our main specification is provided in column (8), which includes the full set of controls and thereby is our Mundlak regression model (see equation 1).The results show that the within estimates are negative (around -0.04) and remain quite stable, although they never become statistically significant.As only 65 practices experience a within variation in    the lack of statistical significance may be due to lack of power.The within estimate indicates that practices experiencing an increase in their list size at a level that exceed their preferred maximum will respond with a reduction of 0.04 services per patient (corresponding to 0.6% of the mean reported in table 1).The between estimates are also negative, which indicates that practices with longer than preferred lists provide fewer services per patient than other practices.The between estimate varies from -0.01 to -0.21, depending on the model specification, but rests on -0.10 in our full model (corresponding to 1.5% of the mean service provision reported in table 1).Appendix table A1 shows the estimates for the covariates.We see that both socioeconomic and morbidity measures statistically significantly explain variation in service provision.As expected, practices with a larger share of deprived patients or patients with specific chronic diseases, on average, provide more services per patient.Especially, the share of patients with COPD is associated with more services.

Table 3: Regression resultslist size above preferred maximum. Dependent variable: contacts (all)
Notes: Djt=1 if the actual list size for practice j in year t is larger than preferred.  ̅ is that average of Djt for practice j.
Charlson 2+ is the share of patients with a Charlson's Comorbidity Index of at least 2. Chronic 2+ is the share of patients with at least two of the following chronic diseases: asthma, dementia, COPD, RA, osteoporosis, schizophrenia, type 1 diabetes, and type 2 diabetes.SES includes low education measured as the share of patients with primary school as highest educational level; low income measured as the share of patients in the first quartile of the disponible income distribution; Retired measured as the share of retired patients; Ethnic Danes measured as the share of patients categorized as ethnic Danes; Transfer income measured as the share of patients between 19-59 that receive transfer income.Chronic diseases include share of patients with; abuse/mental measured as the share of patients that according to the Elixhauser comorbidity index have had either an alcohol abuse, a drug abuse, a psychosis, or a depression within the last 5 years; the share of patients with Dementia, Diabetes, COPD, Osteoporosis, and Schizophrenia.Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1.
Table 4 shows within and between estimates when decomposing the dependent variable into face to face-, email-, telephone-, and annual control consultations as well as preventive home visits.All results in table 4 are based on the Mundlak specification including all covariates.Practices working in a so-called 'list size disequilibrium' may be hypothesized to ration the most time-consuming services (such as annual control consultations and preventive home visits).Table 4 shows that within-estimates are negative for all types of consultations, but none of the estimates are statistically significant.The between-estimates show high reductions in telephone consultation (-0.21) which is around 10% of the sample mean (table 1).All other services show statistically insignificant and positive betweenestimates.The lower level of service provision between exposed practices found in table 3 therefore seems to be driven by fewer telephone consultations.Anecdotal evidence suggests that many practices use telephone consultations in the start of the workday to stratify  patients' inquiries to acute consultations.Our finding of fewer telephone consultation may therefore indicate another organisational structure of practices operating above their preferred maximum list size.Another interpretation could simply be that these practices have less time to conduct telephone consultations.

Heterogeneity analyses
Table 5 and table 6 display heterogenous effects in both the practice and patient dimensions, respectively.All results in table 5 and 6 are based on the Mundlak specification including all covariates.None of the estimates in the practice dimensions are statistically significant.However, table 5 reveals that treated practices with 2-3 GPs and practices operating in areas lacking GPs show the strongest tendencies of rationing.Notes: Only solo practices are included in column 1, group practices of two to three capacities are included in column 2, group practices of at least four capacities are included in column 3, and only clinics in areas that the health authorities define as threatened by GP shortage are included in column 4. All models are adjusted for all covariates (see column 8, Table 3).Djt=1 if the actual list size for practice j in year t is larger than preferred.  ̅ is that average of Djt for practice j.
Table 6 shows the level of service provision for subgroups of patients with various characteristics linked to sociodemographic and disease complexities.These subgroups of patients are defined by having far higher levels of service utilisation in general practice and may therefore be at higher risk of rationing (Hennig-Schmidt et al., 2011, Oxholm et al., 2019).Table A2 in the supplementary material shows that baseline levels of service provision for theses subgroups are three to four times higher than the average of 6.6 services per enlisted patient.Table 6 reveals a couple of statistically significant associations, indicating that some rationing may occur for subgroups of complex patients.Specifically, we see that exposed small group practices (2-3 GPs) provide statistically significantly fewer services to patients with more than one chronic disease that are also on transfer income (-3.6;approximately 19% compared to the mean of 18.9 services (table A2)) and to patients with more than one chronic disease that are also on transfer income and have an abuse or mental disease (-3.3; approximately 16% compared to a mean of 20.9 services (table A2)).The dependent variable is the average number of contact services for the specific patient group.All models adjusted for all covariates.Patient groups are defined as follows: Chronic 2+ is patients with at least two of the following chronic diseases: asthma, dementia, COPD, RA, osteoporosis, schizophrenia, type 1 diabetes, and type 2 diabetes.Charlson 2+ is patients with a Charlson index for at least 2. Transfer income is patients aged 18-59 who receive transfer income.Mental/abuse is patients that according to the Elixhauser comorbidity index have had either an alcohol abuse, a drug abuse, a psychosis, or a depression within the last 5 years.'all' includes patients that both received transfer income and had a mental disorder or an abuse.The number of observations depends on the patient group as practices with less than 5 patients in the specific patient group are excludedsee table A2 for the specific number of clinics *** p<0.01, ** p<0.05, * p<0.1.

Robustness analyses
As shown in figure 1 some practices with a list size above the preferred maximum have quite low deviations from their preferred size.To assess the presence of dose-response effects, we report results of dividing the exposure into practices with a deviation of less than 30 patients per GP and practices with a deviation higher than 30 patients.Table A3 shows that we do not find any evidence of a dose-response relationship.
As a second robustness check, we change the treatment variable to practices with longer lists than what their patient profiles would lead us to expect (  − (  |  ) > 0).For this check, we estimate all the same regressions but replace    with    .Table A4 shows the results of this regression and table A5 and table A6 show the heterogeneity analysis for practice subgroups and patient subgroups, respectively.In general, using the expected list size instead of the preferred list size show comparable results with most estimates being statistically insignificant.However, table A6 shows more statistically significant estimates for associations for subgroups of complex patients, indicating that the increased statistical power in this treatment definition may have an impact.All the statistically significant results for complex patients point in the direction of rationing of services.

Discussion
Many countries, among those the Nordic countries, are currently experiencing an increasing shortage of GPs.Uncovering the consequences of GP shortage is therefore highly policy relevant.The empirical literature is, however, scarce on these consequences.Our study offers a unique contribution by investigating the impact of practices operating with a longer than preferred list on their service provision.We find that practices with a longer than preferred list does not differ substantially on any of the observed patient and practice characteristics.The exception being that they seem to have a slightly more deprived patient population and are also more likely to operate in areas that the health authorities define as threatened by GP shortage.We also find that their service provision to patients is on average similar to other practices.Our subgroup analyses, however, reveal that practices with a longer than preferred list provide fewer services to patients with complex needs.This underprovision of high-need patients aligns with theoretical and laboratory-based studies on physicians' responses to capitation-based payment schemes (Barham and Milliken, 2015, Brosig-Koch et al., 2017, Hennig-Schmidt et al., 2011, Oxholm et al., 2019).Our study contributes to the literature by showing this underprovision in a real world setting where practices face a mixed payment model.
Our study faces some limitations that may explain why we do not find much support of the hypothesis that practices undersupply care when operating with a longer than preferred list.One explanation could be that we lack statistical power as the number of practices who have a reported preferred list size above their actual is limited.Only 67% of practices has reported their preferred list size, indicating a lack of attention of (continuously) reporting this information.This involves the risk that practices' reported preferred list size does not reflect their current preferences.Also, the preferred list size is reported for the entire practice and not the individual GP.As a result, the measure may also suffer from aggregation bias.However, our heterogeneity analyses do not show more statistically significant results for solo practices, indicating that we should not worry about the aggregation.Despite the limitations of the measure of practices' preferred list size, we still consider it a unique measure of work pressure.
Our measures of practices' services are limited by the content of our administrative registers.We can therefore not capture all clinically relevant behaviour.For example, one may hypothesize that practices' underprovision of care may be better reflected in time spent in consultations rather than in the number of consultations.If this is the case, we may be using too crude measures of service provision.More research is therefore needed to investigate underprovision on other dimensions of care than the ones used in this study.The administrative registers also limit our measures of practices' and their patients' characteristics.For example, we cannot capture all dimensions of patients' health statuses.However, the measures we use are commonly used in the literature to characterise patients' need for care (Austin et al., 2015).
Our findings can be generalised to healthcare systems, where GPs are self-employed, operate in a list-based system and receive a mix of capitation and fee-for-services like in the Danish system.A unique feature of the Danish healthcare system is that practices can close their list when reaching 1600 enlisted patients per GP.This feature implies that the risk of practices having a much longer than preferred list is reduced.However, Danish practices may still feel pressure not to close their list due to GP shortage.In other settings where practices cannot close their list for patients, the risk of practices holding a longer list than preferred is higher when there is a GP shortage.We therefore expect that our results provide a conservative estimate of the consequences of GP shortage on service provision.More evidence is therefore needed on these consequences.
Our overall finding that practices operating with a longer than preferred list on average provide the same services to patients as others is reassuring to policymakers.Policymakers should, however, be aware that there may be a cost for high-need patients of the long patient lists.Policymakers should therefore consider incentivising care for these high-need patients.These incentives could be both financial and non-financial (Gneezy andRustichini, 2000, Scott et al., 2011).An example of a financial incentive could be to differentiate practices' payments by their patients' needs, whereas an example of a nonfinancial incentive could be to provide detailed clinical guidelines and performance feedback for treatment of high-need patients (Kongstad et al., 2016, Olsen and Laudicella, 2019, Pulleyblank et al., 2020).
Danish general practices' payments were not differentiated in the period of this study.However, since 2018 their capitation payments have been risk adjusted based on their patients' age, gender, Charlson Comorbidity Index morbidities, and whether the practices operate in an area that the health authorities define as threatened by GP shortage (Danish Regions, 2018).More evidence is, however, needed on the effectiveness of both financial and non-financial incentive schemes on practices' treatment patterns (Anell et al., 2022, Oxholm et al., 2019).

Conclusion
This study finds limited evidence of an association between practices' operating with a longer than preferred patient list and their service provision.However, our subgroup analyses reveal that some rationing on patients with complex needs seem to occur.Policymakers should therefore be aware that there may be underprovision of services to high-need patients when there is a shortage of GPs.

Table 4 : Regression results -list size above preferred maximum. Dependent variable: see column header.
Djt=1 if the actual list size for practice j in year t is larger than preferred.  ̅ is that average of Djt for practice j Standard errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1.All covariates included in all columns. Notes: