T. Iversen, H. Lurås J. of Economic Behavior Org. 43 2000 447–470 449
In Section 2, we present a model that determines a physician’s optimal number of patients and intensity of service provision in the rationed and in the unrationed case. A detailed de-
scription of our data can be found in Section 3. Section 4 describes the estimation procedure and the results of the empirical analysis. Section 5 provides concluding remarks that include
suggestions for further studies.
2. A GP’s optimal number of patients and intensity of service provision
In this section, we model the effect of economic constraints on a GP’s optimal practice profile in a mixed capitation and fee-per-item system. The practice profile is characterised
by the number of patients the GP takes responsibility for, and the volume and composition of services he provides to his patients. We shall assume that the marginal product of health
care on health is a declining function of the volume of services provided. This relation may, for instance, be explained by the fact that the health effect of an initial consultation with a
GP is usually greater than the health effect of the following check-ups. Providing too many health services may even contribute negatively to a patient’s health on the margin. Too many
diagnostic tests increase the probability of false positive results and excessive medication may have considerable adverse effects.
For a number of GP consultations it is not clear what constitutes ‘the right medical treatment’. The reason may be heterogeneity among patients with similar symptoms or
diagnoses. But it may also be due to different opinions among physicians regarding the optimal volume of health services with respect to a patient’s health. For instance, views
among physicians may differ with respect to how often a patient with diabetes or a patient with hypertension should be called for check-ups. Views may also differ on whether a GP
who prescribes antibiotics to a patient should call in the patient for a follow-up consultation next week, or ask the patient to contact him if he feels worse. The intensity of service
provision will on average be higher in the first case than in the second.
5
It is also well documented that referral rates of GPs vary more than can be explained by the composition
of patients. For our purpose, an interesting consequence of the lack of medical standards is that several practice profiles are all regarded as equally satisfactory from a professional
point of view.
6
Fig. 1 describes the range of professional opinions about the relationship between a health service, s
i
, and its marginal health effect, hs
i
. The downward sloping line to the left expresses the most pessimistic view, while the downward sloping line to the right expresses
the most optimistic view about the marginal effect of health services. The figure describes an interval, [α
i
, α
i
+ β
i
], where the marginal effect of health services is not documented to be different from zero.
7
Medical practice variation and the lack of medical standards are arguments for the existence of this interval. Other things equal, professional consensus
about treatment profiles contributes to narrowing the interval.
5
A thoughtful discussion of the various aspects of uncertainty in knowledge about the efficacy of medical interventions is given by Phelps 1992.
6
For an introduction to the literature on medical practice variation, see Andersen and Mooney 1990.
7
In the health economics literature this interval is often referred to as ‘flat of the curve medicine’, see for instance Enthoven 1980.
450 T. Iversen, H. Lurås J. of Economic Behavior Org. 43 2000 447–470
Fig. 1. The range of professional opinions about the marginal effect of health services.
We denote the number of health services in excess of α
i
as k
i
. To simplify the anal- ysis, we shall assume only two health services. Furthermore, α
i
is assumed to depend on k
j
, i6=j, such that α
i
= α
i
k
j
. For instance, if service no. 1 is consultations in the physician’s office and service no. 2 is consultations by telephone, it seems reasonable that
∂α
1
k
2
∂k
2
0. On the other hand, if service no. 1 is medication requiring monitoring, we may have ∂α
1
k
2
∂k
2
0. The value of α
i
and β
i
may vary between groups of patients. The number of episodes of illness during a certain period, as well as the marginal benefit from health services during an
episode of illness, may differ. For instance, elderly persons with chronic diseases may need more health services to stay healthy than young people. We introduce a parameter, θ , to
account for variation in need according to patient group. θ is considered to be exogenous to the individual GP.
8
We then have α
i
= α
i
θ , k
j
and β
i
= β
i
θ , where the partial derivatives with respect to θ are positive. We shall assume that the distribution of θ describes the
physician’s population of patients. θ has the probability density function fθ , where θ ∈0, ∞
. Fθ is the cumulative distribution function. We shall assume that the physician can influence the number of patients n he takes care off. The distribution of patients according
to groups is, however, assumed to be exogenous to the physician. We assume that a physician decides the quantity of health services provided to each of
his patients.
9
We further assume that a physician has lexicographic preferences concerning his patients’ health and his own income and leisure. This assumption implies that a patient’s
health is never balanced against the GP’s income or leisure. Health services are then provided until the marginal health effect is equal to zero. This assumption simplifies the formal
reasoning considerably and it is also in accordance with the medical profession’s own
8
Since we have a one-period model, it is reasonable to assume that θ is exogenous. In a two-period model θ in this period may, however, depend on the volume of health services provided in the previous period.
9
The number of patient-initiated contacts is assumed to be included in α
i
.
T. Iversen, H. Lurås J. of Economic Behavior Org. 43 2000 447–470 451
assertion. A relaxation of this assumption would imply that the effect of economic incentives is strengthened.
We assume a quasilinear objective function formulated in monetary terms as V=c+vl, where c is consumption all income is consumed and l is leisure. vl is assumed to be strictly
concave with v
′
l0. The physician’s decision problem may now formally be expressed as max c
k
1
,k
2
,n
+ vl = w + qn + n
Z
∞
{ p
1
[α
1
θ, k
2
+ k
1
] +
p
2
[α
2
θ, k
1
+ k
2
]}f θ dθ +
v[T − n Z
∞
{ t
1
[α
1
θ, k
2
+ k
1
] + t
2
[α
2
θ, k
1
+ k
2
]}f θ dθ ] 1
such that i
0 ≤ k
1
≤ β
1
θ ii
0 ≤ k
2
≤ β
2
θ iii
0 n ≤ ¯ n
where w is a fixed salary or practice allowance, q is a capitation payment per person on the physician’s list of patients, p
i
is the fee per item of health service or equivalently, a fee for service no. i i=1, 2,
10
t
i
is the time input of providing health service no. i, T is the total time endowment and ¯
n is a constraint on the number of patients. All of these variables are assumed to be exogenous to the physician.
11
i and ii say that the number of services provided to a patient must be within the interval where the marginal effect of services on
health is not documented to be different from zero. iii says that the number of patients is positive, but physicians may experience a shortage of patients. We shall only consider cases
where n is positive.
The maximisation problem is analysed by means of concave programming.
12
Because the empirical part of the paper employs data from a mixed capitation and fee-per-item
system, we shall concentrate on the model’s predictions under this system.
13
When q0, p
1
0, p
2
0, w=0 and ‘a’ is the net fee per time unit of service no. 2 relative to service no. 1 we have the following results.
In a combined capitation and fee-per-item system: •
The minimum volume of both services is provided when the income per time unit of providing service no. 1 is equal to the income per time unit of providing service no. 2
a=1 and in an interval around a=1. •
Outside this interval: when the income per time unit of providing service no. 1 is larger smaller than the income per time unit of providing service no. 2, the maximum mini-
mum volume of service no. 1 and the minimum maximum volume of service no. 2 are provided.
10
p
i
may be interpreted as a net fee, i.e. the gross fee minus the physician’s costs for employed personnel and other inputs except own time.
11
Assuming t
i
to be exogenous is not quite realistic, but it simplifies the analysis and plays no substantial role for our conclusions.
12
Details can be found in Appendix A.
13
Results for other payment systems can be found in Appendix B, available from the authors.
452 T. Iversen, H. Lurås J. of Economic Behavior Org. 43 2000 447–470
• When a shortage of patients occurs, even the less profitable service may be provided in
excess of the minimum volume. The intuition behind the result is: in a pure fee-per-item system the relative fee per time
unit a determines the optimal number of each service. If a6=1, the least paid service will be provided at the minimum volume k
i
= 0, while the best paid service will be provided
at the maximum volume k
j
= β
j
. On the other hand, in a pure capitation system, where the list of patients is the only income-generating component for the GP, the volume of both
services is chosen equal to the lower constraint. In a combined payment system, the size of the capitation component relative to the fee-for-service component determines the optimal
service provision. Even if a6=1, extra provision of the best paid service has an opportunity cost of reduced income from capitation. If this opportunity cost is ‘large enough’ relative
to the fee-for-service component, the provision of both services is kept to the minimum k
1
= k
2
= 0. With a shortage of patients, the opportunity cost declines and hence, the service
provision is likely to increase. The solution is shown in Fig. 2. In Appendix A, it is shown that k
1,
k
2
and n are homogenous of degree zero in p
1
, p
2
and q. In Fig. 2, the curve that illustrates the margin between the k
1
= k
2
= 0 and k
1
= β
1,
k
2
= 0 is downward sloping because a decline in ‘a’ makes it more profitable to increase the
provision of service no. 1. Hence, an increase in q is necessary to restore the indifference. By similar reasoning, it follows that the curve showing the margin between k
1
= k
2
= 0 and
k
1
= 0, k
2
= β
2
is upward sloping. Fig. 2 also illustrates that the introduction of a capitation component implies that k
1
= k
2
= even when a6=1 is in the intervala, ¯a. The result illustrates a point emphasised by Newhouse
1992, 1996; because of the practical difficulties of fine-tuning a single payment component, a mixture of payment components is more robust in order to achieve health policy objectives.
Let us finally consider the feasible solutions when there is a shortage of patients, i.e. n= ¯ n.
Let µ be the Lagrange parameter related to the third constraint in 1, expressing the increase in utility of a marginal loosening of the constraint, i.e. a marginal increase in the number of
patients. Hence, µ0 when n= ¯ n. The magnitude of µ relative to the per capita component
q plays an important role in the constrained case. If µq, the increase in utility is less
Fig. 2. Regions of provision of health services as the capitation fee and services fees vary.
T. Iversen, H. Lurås J. of Economic Behavior Org. 43 2000 447–470 453
than the capitation fee. This means that the marginal utility from increasing the volume of services with the optimal composition of services is negative: the marginal value of leisure
is larger than the income from marginal service provision. Then, the number of services per patient in the constrained case is equal to the number of services in the unconstrained
case. If µq, the income from marginal service provision with an optimal composition of services is larger than the marginal value of leisure. The number of services per patient in
the constrained case is then larger than the number of services in the unconstrained case. The most profitable service is certainly offered at the maximum level, and if the shortage
of patients hurts enough, even the less profitable service may be offered in excess of the minimum volume. We are not able to distinguish empirically between GPs according to the
relation between µ and q. However, if µq for at least one GP in our sample, we will find an effect of rationing on the volume of services provided. In Appendix B, we show that if
µq, the maximum number of both services is provided even with a=1. The interpretation of µ=q should now be obvious.
From the model’s result the following testable hypotheses follow directly. •
A GP’s income from fees per listed person is expected to be higher in the constrained case a shortage of patients than in the unconstrained case, because more services per
person are expected to be provided. •
If the constraint hurts enough, even the less profitable services are expected to be provided in excess of the minimum volume in the constrained case.
We shall also take into account an effect of patient composition that follows directly from one of the model’s assumptions.
• A GP’s income per listed person increases when the composition of the list shifts towards
a population with a greater need for health services a negative shift in Fθ .
3. Description of the data
