General motivation and model layout
In theory, the spatial differences of physicians can be explained by demand-side as well as by supply-side factors. The regional distribution of physicians is usually seen as a market failure due to physician-induced demand (see Richardson and Peacock [
25], van Dijk et al. [
26]) calling for governmental intervention. Our model does not assume that physicians are able to induce demand for their services, so that the physician density in a given region is determined by the patients’ demand and the regional preferences of the physician (see for physician regional preferences Matsumoto et al. [
27]). A more detailed overview of supply-side factors from the literature is given and examined in the last part of this section.
The theoretical model has four parts. The first part presents a model of the demand-side. It is a simple model of a representative individual who demands services by general practitioners (family doctors) and specialists, as we are also interested in the composition of the physician in urban and rural areas. In the second part, a consumption time function is introduced. Our hypothesis is that search, traveling, and waiting time play an important role for the consumer for his or her decision to consult a doctor. The higher the time cost to consume a doctor, the lower is the demand. With this approach, we incorporate a well-known argument already raised by the seminar article of Phelps and Newhouse [
28].
The third part presents a model for physicians deciding to settle in a certain region. We assume that physicians like other professionals are interested in income which is the revenues deducted by the cost of running the office. In the fourth and final part, we derive the regional equilibrium to determine the physician-population ratio in urban and rural areas. In this part, we also introduce a factor covering the regional preferences of physicians. We assume that a physician who settles in a more preferred region is willing to sacrifice for this part of his or her income. A regional equilibrium is reached, if every physician belonging to a certain specialty earns the same income weighted by the regional preference factor.
In our model, we assume two regions, an urban (u) and a rural one (r). Both regions have the same population (n = nu = nr) but differ in their size measured in square miles or square kilometers m. The size of the rural region is mr and of the urban area mu (mu < mr). Thus, the population density n/m is higher in u than in r. The regional physician density is GP/m and specialist (SP)/m, where GP and SP are the number of practitioners and specialists, respectively. The number of physicians per capita is GP/n and SP/n.
It is assumed that there is no regional difference in morbidity or demographics, with all consumers and physicians having the same preferences and physicians—as mentioned above—not being able to induce demand. There are no quality differences between physicians, and the services provided are homogeneous.
Demand for physicians’ services
We consider a representative individual that has a fixed income of
y, which is spent solely for consumption. The expenditure for consumption—except physician services—is
Y. The payment for health insurance is payroll tax on income
y at a tax rate
b. A and
S are the services of general practitioners and specialists, respectively.
p is the price for those services. That means
p
A
is the price for one unit of services by a GP.
p
S
is the price for consulting a SP.
á represents the co-insurance rate, so that the patient has out-of-pocket payments of
áp
A
or
áp
S
for a service unit of a GP or a SP. Thus, the monetary constraint for consumption is given by
$$ y = Y + b\ y + \acute{a}\kern0.5em {p}_A\ A + \acute{a}\kern0.5em {p}_S\ S $$
(1)
The consumer has also a time constraint with his or her time budget being
l. This is distributed on consumption time
L and for the time consulting a doctor. The time coefficients
t
i
(with
i =
A,
S) are the time needed to see a physician, including travel time, waiting time, and consultation time.
$$ l = L + {t}_A\ A + {t}_S\ S $$
(2)
To sum up, the consumer spends his or her money and time either for physician services or for other consumption. The resources (money and time) taken for other consumptions are expressed by
$$ Z=Y + \varrho L $$
(3)
where
ϱ denotes the individual’s opportunity cost of time. In a perfect world with flexible work time, it would correspond to the wage rate (see, for example, Frank [
29]).
To finalize the demand model, it is assumed that the individual maximizes a utility function. To receive concrete results, this function is specified. We assume a Cobb-Douglas-type utility function:
$$ U={A}^a{S}^s{Z}^z,\ s>a $$
(4)
This type of function has plausible properties as described in the literature.
s >
a means that specialists’ activities are preferred over GPs’ activities, but GPs’ activities cannot be fully substituted by specialists’ activities and vice versa. Obviously, both types of services are needed, but a partial substitution is possible. Our assumption that patients may substitute one physician type for another if needed is in line with general observations and empirical studies. McLeod [
30] showed, with data from the Canadian Community Health Survey and the Ontario Health Insurance Program, that a shortage in the supply of one physician type can result in an increase in the use of other physician types.
Our model does not cover the fact that the demand for specialists’ services can be the result of referrals of general practitioners. In some countries, like Germany, Austria, and Switzerland, patients may consult a specialist directly without a referral from a family doctor. In other countries, like Great Britain, a family doctor has to be consulted first before going to a specialist. In some countries, in addition, specialists do only work in hospitals. Gächter et al. [
31] have shown for Austria that referrals from GPs to SPs play a role and that GPs and SPs collaborate with each other, so that both markets are interrelated.
Inserting the monetary constraint (1) and the time constraint (2) in (3) leads to
$$ 0 = Z\ \hbox{--}\ y + b\ y + \acute{a}\ {p}_A\ A + \acute{a}\ {p}_S\ S - \varrho\ l + \varrho\ {t}_A\ A + \varrho\ {t}_S\ S $$
(5)
The (representative) individual maximizes his or her utility function (6) subject to (5), whereby
Q is the maximization operator and
λ a Lagrange multiplier, which is used to maximize an objective function under constraints.
$$ \max\ Q = {A}^a{S}^s{Z}^z + \lambda \left[Z\ \hbox{--}\ y + by + \acute{a}{p}_AA + \acute{a}{p}_SS - \varrho l + \varrho {t}_AA + \varrho {t}_SS\right] $$
(6)
Differentiating (6) with respect to
A,
S,
Z, and
λ leads to the first-order conditions for an optimal consumption plan. If we divide these conditions pairwise, we obtain
$$ \frac{S}{A} = \frac{\left(\acute{a}{p}_A+\varrho\ {t}_A\right)}{\left(\acute{a}{p}_S+\varrho\ {t}_S\right)}\ \frac{s}{a} $$
(7)
$$ \frac{Z}{S}=\left(\acute{a}{p}_S+\varrho {t}_S\right)\ \frac{z}{s} $$
(8)
$$ \frac{Z}{A}=\left(\acute{a}{p}_A+\varrho {t}_A\right)\frac{z}{a} $$
(9)
As the model is constructed, the relative demand for services of GPs and SPs and the demand for other consumptions depend on the insurance coverage, the price of other consumptions, the time needed to consume physician services, and the preferences expressed by the coefficients in the utility function.
If, for instance, the time needed to consume specialist services, t
s
, is higher or the preference for those services is lower, the demand for services of general practitioners A will be higher relative to S and Z in the utility-maximizing consumption plan.
In the next step, we solve the equations (7) to (9) for
S and
Z, respectively. This allows us to replace
S and
Z in (5) by (7) to (9), so that only the variable
A is left and
S and
Z have been replaced. We obtain
$$ \left(\acute{a}{p}_A+\varrho {t}_A\right)\ \frac{z}{a}\ A-y\left(1-b\right)-l+\left(\acute{a}{p}_A+\varrho {t}_A\right)A+\left(\acute{a}{p}_S+\varrho {t}_S\right)\ \frac{\left(\acute{a}{p}_A+\varrho {t}_A\right)}{\left(\acute{a}{p}_S+\varrho {t}_S\right)}\ \frac{s}{a}\ A=0. $$
(10)
This leads after reorganization to
$$ A = \frac{a\ \left(y\left(1-b\right)+\varrho l\right)}{\left(\acute{a}{p}_A+\varrho {t}_A\right)\left(a+z+s\right)}. $$
(11)
In the same way, we can derive from (7) to (10) the equation for
S:
$$ S = \frac{s\ \left(y\left(1-b\right)+\varrho l\right)}{\left(\acute{a}{p}_S+\varrho {t}_S\right)\left(a+z+s\right)}. $$
(12)
By multiplying the number of inhabitants in each region with the demand for physician services, we receive the demand for the whole region, i.e., nAu, nAr, nSu, and nSr. This demand is dependent on the number of inhabitants (which is assumed to be n and the same in each region); the preferences expressed by a, s, and z; the net income (1 − b)y, the monetary cost áp
A
(which is larger than 0 if the co-insurance á is positive); and the time cost ϱt
S
to consume the services (which are different in the regions due to difference in traveling time). If we divide nAu, nAr, nSu, and nSr by the number of physicians, e.g., GPu, GPr, SPu, and SPr, we receive the average number of services provided by each physician.
Time cost
The time cost to consult a physician includes travel time, waiting time, and time of consultation. Traveling time and by this time cost decrease with the average regional distance between physicians’ practices. Whereas in urban areas the nearest doctor’s office can be reached in just a few minutes, travel times (hence time cost) in rural areas are more considerable.
Following this argument, we assume the following time cost function:
$$ {t}_{A_i} = {\left(\frac{{\mathrm{GP}}_i}{m_i}\right)}^{-{q}_A},\ \mathrm{with}\ i = \left\{\mathrm{u},\mathrm{r}\right\};\ 0 < {q}_j < 1;\ {q}_S > {q}_A $$
(13)
$$ {t}_{S_i} = {\left(\frac{{\mathrm{SP}}_i}{m_i}\right)}^{-{q}_S} $$
(14)
where GP
i
stands for the number of general practitioners, SP
i
for the number of specialists, and i for the region. mi measures the size of the region. Although the population is assumed to be the same in both regions, the rural region is larger than the urban one. q
A
and q
S
are the time cost coefficients. If q
j
(j = A,S) is 0, the time cost does not vary with the physician density. If q
j
is 1, the time cost varies proportional with the physician density. We assume q
j
to be between 0 and 1.
The explanation for the chosen time cost equation is as follows: The time cost is dependent on various factors, but also on the regional physician density, because the average travel time increases for the patients if there are fewer physician offices in the region. If the number of physicians goes up in a certain region, the time cost will decrease and q is assumed to be smaller than 1, because the traveling time is only one component of the time needed to visit a physician’s office. The patient also has to invest time to search for the right physician, needs the waiting time in the office, and the time for the treatment itself. q smaller than 1 also implies that the demand for physician services does not grow faster than the number of physicians, i.e., the time-cost-physician elasticity is smaller than 1.
It is plausible that it takes more time to find an appropriate specialist, due to the increasing variety of specialists. Because the scarcity of certain specialists in large areas also increases travel time, we assume q
A
< q
S
.
Physician behavior
The third part of the model specifies physician behavior. For simplicity, we assume that physicians have some regional preferences, which we will introduce in the next section, but are interested in their income. So physicians move to that region where they receive a higher income.
The prices for the physicians’ services are the same in both regions but differ between the types of physicians, e.g., they are
p
A
and p
S. Assuming constant marginal cost and that all physicians have the same number of patients in a given region, the income of a GP and a SP in a rural or urban region is as follows, where
Y
Ai
and
Y
Si
with
i = u, r are the income of a physician in the urban and the rural region, respectively, and
c
A
and
c
S
are the marginal cost for one service unit of a GP or a SP, respectively:
$$ {Y}_{A_i}=\left({p}_A-{c}_A\right)\ {A}_in/{\mathrm{GP}}_i\ \mathrm{with}\ i=\mathrm{u},\mathrm{r} $$
(15)
$$ {Y}_{S_i}=\left({p}_S-{c}_S\right)\ {S}_in/{\mathrm{GP}}_i\ \mathrm{with}\ i=\mathrm{u},\mathrm{r} $$
(16)
with j = {GP, SP}, i = {u, r}.
If physicians are free to choose where to open their practice, in a state of equilibrium, physicians’ incomes are the same in both regions:
$$ {Y}_{A_{\mathrm{r}}} = w\ {Y}_{A_{\mathrm{u}}},\ w\ \ge\ 1 $$
(17)
and
$$ {Y}_{S_{\mathrm{r}}} = w\ {Y}_{S_{\mathrm{u}}},\ w\ \ge\ 1 $$
(18)
w expresses the regional preference of physicians. If w is 1, physicians have no regional preferences. If it is greater than 1, they prefer urban areas, i.e., they are willing to sacrifice part of their income to live in a preferred region. Physicians are producers and as well as consumers, and they choose to go to places for many reasons. One reason is certainly to have enough patients. But as a consumer, other reasons are important, like attractiveness, cost of living, and personal tasks.
It is certainly a rough simplification to assume that urban areas are always more attractive for physicians than rural ones. For instance, Matsumoto et al. [
23] have shown in a study on Japanese physicians that the attractiveness of a municipality depends on the amenities of urban life which seems to be more highly correlated with the number of “daytime population” and “service industry population” than total population. Erus and Bilir [
32] have published recently results from a Turkish study, which shows that after the regulation was lifted that young doctors have to go first in underserved regions; socio-economic conditions of a region became a significant determinant of availability of specialists. So it is important to detect the factors or proxies for the attractiveness of a region.
Regional equilibrium
In the next, final part of the model, we bring together the three previous parts of the model and derive some conclusions. We insert (11) to (16) in (17) and (18) and receive for
á = 0 the following equations:
$$ \frac{{\mathrm{GP}}_{\mathrm{u}}}{{\mathrm{GP}}_{\mathrm{r}}}={\left(\frac{m_{\mathrm{r}}}{m_{\mathrm{u}}}\right)}^{\frac{q_A}{1-{q}_A}}\ {w}^{\frac{1}{1-{q}_A}} $$
(19)
$$ \frac{{\mathrm{SP}}_{\mathrm{u}}}{{\mathrm{SP}}_{\mathrm{r}}}={\left(\frac{m_{\mathrm{r}}}{m_{\mathrm{u}}}\right)}^{\frac{q_S}{1-{q}_S}}\ {w}^{\frac{1}{1-{q}_S}} $$
(20)
(19) and (20) show that the number of physicians and hence the physician density is larger in urban than in rural areas, because the right-hand side is greater than 1. The urban-rural discrepancy increases if the regional preference for urban areas increases.
Dividing (20) by (19) yields to
$$ \frac{{\mathrm{SP}}_{\mathrm{u}}}{{\mathrm{GP}}_{\mathrm{u}}}=\frac{{\mathrm{SP}}_{\mathrm{r}}}{{\mathrm{GP}}_{\mathrm{r}}}\ {\left(\frac{m_{\mathrm{r}}}{m_{\mathrm{u}}}\right)}^{\frac{q_S-{q}_A}{\left(1-{q}_A\right)\left(1-{q}_S\right)}} $$
(21)
Obviously, the specialist-general practitioner relationship is higher in urban than in rural areas if q
A
< q
s
.
Our model leads to four hypotheses:
1.
Regional preferences of physicians lead to differences in regional physician-population ratio, even if physicians are not able to induce demand for their services. In our model, the preferences of physicians for urban areas are reflected by the parameter w. We will introduce a number of proxies for the attractiveness of a region in our empirical analysis.
2.
If physicians have no regional preference, it is also plausible that the physician-population ratio for each specialist group is higher in regions with a high-population density than in rural areas. If traveling and waiting time plays a role for the demand for physician services, and these time costs are high in a given region because the physician-population ratio is low, the demand will increase, if the physician-population ratio increases.
3.
Not only the absolute number of specialists but also the GP-specialist ratio is higher in urban than in rural areas, if search and travel time costs are higher for specialists. As the two first propositions derived from the model are straightforward, the third proposition needs to be explained. The reason is the heterogeneity of specialists compared with general practitioners. The more different types of specialists exist, the higher are the time costs to find the right doctor for the patient’s specific health problem. An increase in the number of physicians will increase the demand for their services due to the reduced time cost. But this effect is larger for SPs than for GPs.
4.
The higher the level of insurance coverage of physician services or the lower the co-payments, the higher the regional inequality of outpatient care. This can be derived directly from our model. If, for instance, á is set equal to 1 instead of 0, the monetary cost increases. By this, the relative importance of the time cost decreases.
Hypotheses 1 and 2 make clear that it is an empirical question whether differences in physician-population ratio are really a sign of market failure. The question is whether and to which extent demand differences or regional preferences of the physicians are associated with the differences in physician-population ratio. One factor causing demand differences is the time cost associated with medical treatment. Other factors are morbidity differences. While the regional physician-population ratio and the size of a region are known, the challenge of an empirical study will be to detect the right proxies for the demand for medical services and the attractiveness of a region.
The model is simple and, not surprisingly, it has a number of limitations. One limitation of the model is the assumption that physicians are income maximizers. If physicians behave in a purely altruistic manner, then the model does not correctly describe physician behavior. However, our hypothesis that monetary incentives are one of the main drivers for the decision of setting up a practice is confirmed by Günther et al. [
33]. In a survey conducted among 14 939 German practicing non-postgraduate physicians, the authors used a discrete-choice experiment to weight the attributes of hypothetical locations for practices. Income was weighted with the highest utility weight.
Physician preferences and supplementing factors
However, other parameters also play a role in the decision for a certain location, and it is important for the empirical validation of our model to find appropriate proxies not only for demand but also for the supply side from the literature. Also in the study of Günther et al. [
33], the authors found the attributes
professional cooperation,
leisure activities,
career opportunities of the partner, and
availability of child care followed the utility weight of income in descending order from qualitative interviews. Thereby, the last three attributes were equalized by the driving distance. This survey also included a questionnaire asking physicians about the importance of 18 items on their decision for a practice location. Using factor and regression analysis on the results of the questionnaire, Roick et al. [
34] found financial incentives to be less important than a
positive environment for the family and
occupational duties. On the other hand,
possibilities for cooperation,
conditions of service, and
quality of life in their living area in descending order are of lower relevance.
The study from Kazanjian and Pagliccia [
35] used a very similar methodological framework as Roick et al. [
34] to retrospectively analyze Canadian physicians’ choice of practice location in 1989. The authors focused on physicians’ satisfaction with the location of their current practice on the basis of 41 items as well as on persons and events influencing the decision for this location and thereby differentiate between physicians practicing in rural and urban areas. Physicians’ spouses, the desire to raise a family in an environment similar to that of their own childhood, peers, and friends have the highest influence on the location of physicians’ practices. Nevertheless, income and other location factors are only part of the equation when analyzing satisfaction with the current location of practice and therefore do not allow any conclusions to be drawn between them and the primary choice of residence.
An empirical model from Breyer et al. [
36] analyzed the determinants of utilizing physician services and the supply side in Germany. It showed that the determinants of physician supply are explained by per capita expenditure for physician services, population density, gross domestic product (GDP) of the region, hospital beds per 1000 inhabitants, and overnight stays in hotels. The last two parameters are assumed to cover the distance to the physicians’ training location, i.e., university hospital, and the cultural and recreational appeal of a region. In the regression analysis, only expenditure for physicians, hospital bed density, and overnight stays in hospitals showed a significant association with the physician density.
A comprehensive analysis has also been conducted by Kistemann and Schröer [
4]. In this study, the target and actual numbers of GPs and different specialists are compared in a German district on the postal-code level as a function of the portion of inhabitants of high socioeconomic status. In an additional postal survey, physicians were asked about the importance of economic and personal factors on the decision for their current practice location. The authors found an equal number of non-economic and economic factors being rated “important”.
As these publications reveal, economic incentives, measured either directly over the expenditure for physician services as in Breyer et al. [
36] or indirectly over the population density as in our model, are not exclusively responsible for physicians’ decision regarding the location of their practice. Instead, the professional environment for the physician, a labor market for the physicians’ spouses, the accessibility of cultural and recreational activities, and the attractiveness of a district are contributing factors to this decision. The following empirical analysis seeks to examine to what extent the propositions of our model can be found in German outpatient care and which other spatial factors are associated with the number of GPs and specialists.