1 Introduction
Confronted with high demands for health care and a limited budget, most countries need to prioritize health care expenditures among their population, which becomes an important aspect of 6health care management. In his seminal work, Arrow
[1] pointed out that health care relates mainly to two sources of risk, i.e., the existence of uncertainties in the severity of disease and in the effectiveness of medical treatment. Dardanoni and Wagstaff
[2] analyzed the impact on the demand for medical care of each of the disease and treatment uncertainties
1. In a stimulating paper, Hoel
[5] proposed the issue of priority in health care expenditures, and demonstrates that under risk aversion with respect to health outcome, the allocation of health resources should be directed to health conditions for which the expected outcomes are below average
2. Bui, et al.
[6] indicated that besides risk aversion, the degree of absolute prudence (the convexity of the marginal utility function) also matters to increase the budget allocated to the more severe disease. In fact, Hoel
[5] and Bui, et al.
[6] examined the health care allocation problem by introducing an uncertainty on health outcome. Recently, Courbage and Rey
[7] considered very general types of changes in disease risk and treatment risk, respectively. They conclude that, (a) when only the severity of illness is uncertain, more resources should be allocated to the patients who are more at disease risk in the sense of
th-degree risk increase of Ekern
[8] if, and only if,
; (b) when only the effectiveness of health care is uncertain, more resources should be allocated to the patients who are more at treatment risk in the sense of
th-degree risk increase if, and only if, the measure of relative
th-degree of risk aversion exceeds
, i.e.,
3; (c) under inequity aversion, the priority setting results (a) and (b) are not modified in the sense that more resources should be allocated to the patients at greater risk under the same preference conditions.
1Actually, they described variable uncertainty by mean preserving spread of Rothschild and Stiglitz
[3], whereas Leshno and Levy
[4] characterize change in risk by stochastic dominance in medical decision making.
2As in [
5], health outcome is simply defined as the number of life years that the individual.
3Chiu, et al.
[9], Denuit and Rey
[10] and Wang and Li
[11] gave an interpretation for the value of
th-degree relative risk aversion by a particular class of lottery preferences.
In spite of the literature has provided some fruitful findings, but there are several important issues left. To begin with, it is worth noticing that previous health care priority models don't consider a disease uncertainty and a treatment uncertainty at the same time. In many real-world circumstances, a system change always involves some contemporaneous uncertainties. For this reason, when the two types of uncertainty are introduced simultaneously, the analysis of health care priority is quite valuable. Furthermore, the attentions of most researchers are paid to the impact of a single independent risk on the allocation of health care. In two-risk framework, it is significant that the correlation structure between two risks enter into the exploration of health care allocation. Lastly, it is interesting to examine whether the inequality aversion of the social planner affect health care allocation in the model of two risks or not. The vital problems above-mentioned lack discussion in previous literature, the main purpose of this paper is to fill this gap.
To conduct our work, we rely on the recent research in Courbage and Rey
[7] modeling the problem of priority in health care in one-risk framework. By virtue of the mature technique to study precautionary saving (e.g., Courbage and Rey
[12], Denuit, et al.
[13], Li
[14], Menegatti
[15]), this paper examines the health care priority issue in the simultaneous presence of disease and treatment uncertainties. More precisely, when the disease and treatment risks are independent, we provide social preference conditions under which an introduction of a disease or treatment uncertainty leads more health care resources to be allocated to higher risk population. But when the two risks are dependent, in order to maintain health care priority, more restrictive conditions on preferences are required. In addition, when the two risks are small or positively quadrant dependent, we point out stronger preference structure to guarantee that the joint impact of two uncertainties on the resource allocation for populations at higher risk is positive. Considering an equity weighting function, the role of attitude towards inequity on health care allocation is analyzed. We show that the priority setting conclusions in this paper are driven by the dependence between two risks, the sign of the successive third order derivative of the utility function and the intensity measure of partial relative prudence.
The remainder of the paper is organized as follows. Section 2 outlines our model and gives the necessary and sufficient conditions for priority setting in health care. Section 3 demonstrates our comparative static results relating the optimal allocation of health care resources in the framework with multiple risks. Section 4 addresses the impact on health care allocation of aversion towards inequity. Section 5 briefly concludes.
2 The Model
As in Arrow
[1] and Dardanoni and Wagstaff
[2], we consider two sources of uncertainty: One is the variability of the effectiveness of medical treatment or the productivity of health care, denoted by
, where
represents its stochastic component and
represents its certain one with
and
for all levels of medical care
, i.e., more investment in medical care will improve the health status of the patient at a decreasing rate; the other is the randomness of the basic level of health (health condition), denoted by
, which reflects the severity of the possible disease
4. It is assumed that health could be quantified, for instance, via quality-adjusted life-years
. The uncertain health level of the individual,
, is of the form
5:
4Random variables such as and are assumed to be positive and bounded above.
5In this paper, the disease and treatment risks exist at the same time and appear in additive and multiplicative form, respectively.
Following Bui, et al.
[6], Courbage and Rey
[7], Hoel
[5], we consider a social planner whose preference is represented by a social von Neumann-Morgenstern utility function
defined over health
with
and
, i.e., she is non-satisfied and risk-averse in health
6. The social planner who subjects to a fixed budget
has to allocate health care resources across a population. The population is composed of two types of individual and the proportion of individual of type-
is denoted by
, with naturally
. The health level for the type-
patients is given by
. The levels of health care expenditure
and
, respectively allocated to type-
and type-
patients, are chosen by the social planner in order to maximize expected social welfare. The optimization problem is then expressed by the Lagrangian function
:
6Throughout the paper, random variables have a tilde whereas their realizations do not.
where stands for the expectation operator. If the interior solutions prevail, the corresponding first order conditions (FOC) yield:
in which the levels and denote the optimal allocations of type- and type- patients, respectively. For our purpose, it is useful to define a bivariate function:
We have
and
in which for a required couple of integers , denotes the th partial derivative of the function . Using a second order two-variable Taylor expansion around the point , in which and , the bivariate function is approximated as
We are able to state the following lemma.
Lemma 1 Suppose that two types of patient confront uncertainties in the effectiveness of health care and the severity of illness and , respectively. The social planner should allocate more less resources to type- patients than to type- patients, i.e., if, and only if, .
Proof Denote:
Since , and , we obtain
That is, is decreasing in . From (11) and (12), the optimal expenditures and satisfy if, and only if,
From (3) and (13), we get
By (4), (14) can be written as
Therefore, in the presence of two types of risk, the social planner will invest more (less) in health care for type- individuals than for type- individuals, i.e., if, and only if, the inequality (15) holds.
Lemma 1 reveals that a decision maker who prioritizes two types of patient in population corresponds to another decision maker who exhibits preference over two bivariate risks. To analyze the allocation of health care resources, let us impose different assumptions on the disease and treatment risks of the two types of patient7:
7To make a distinction between the two types of population, we simply require that type-1 patients have more disease or treatment risk than type-2 patients in the sense of mean preserving increase in risk of [
3], which is in line with an introduction of a disease or treatment uncertainty.
(a) Two types of individual face the same treatment risk and differ in the severity of disease ( and );
(b) Two types of individual face the same disease risk and differ in the effectiveness of treatment ( and );
(c) Two types of individual differ both in the severity of disease and the effectiveness of treatment ( and ).
For case (a), we examine the marginal effect of the disease uncertainty on health care priority in the presence of a multiplicative treatment risk; For case (b), we explore the marginal effect of the treatment uncertainty on health care priority taking into account the presence of an additive disease risk; For case (c), we study the joint effect on health care priority of the contemporaneous presence of the two uncertainties.
3 Uncertainties and Priority in Health Care
Let us start with our discussion for the situation where the two types of population only differ in the uncertainty about one of the severity of the illness and the effectiveness of health care. We first investigate the impact of one uncertainty on health care allocation in the presence of other risk. When the type- individuals only confront the identical treatment risk, we replace by in (2). Let and be the optimal expenditures on type- and type- patients, respectively. It is apparent that the allocation of health care to the two types could be clarified by comparing with .
Similarly, when the type- individuals only confront the identical disease risk, we replace with in (2). In this context, the optimal expenditures of type- and type- patients are and , respectively. In order to make clear the health care allocation of the two types, the comparison between and is needed.
When the disease and treatment risks are mutually independent, on the basis of an approach to investigate precautionary saving in the setting of independent risks of Courbage and Rey
[12], we capture the result as follows.
Proposition 1 (a) Suppose that two types of patient only differ in the severity of illness, for which the disease risks are and , respectively. For independent risks, the social planner should allocate more resources to type- patients than to type- patients, i.e., if, and only if, .
(b) Suppose that two types of patient only differ in the effectiveness of health care, for which the treatment risks are and , respectively. For independent risks, the social planner should allocate more resources to type- patients than to type- patients, i.e., if, and only if, , .
Proof From Lemma 1, it is obtained that . When the two risks are independent, for any bivariate function and any , it is easily known that . Thanks to (7) and (8), we get
and
Thereby, we have if, and only if, the inequality (16) ((17)) holds.
As indicated by Kimball
[16], the decision maker with convex marginal utility function, i.e.,
, with
, displays prudence and she has positive precautionary saving motive. The partial relative prudence measure is defined as
, which is used to influence various economic behaviors
8. Given that the disease and treatment risks are independent. The implication of Proposition 1 (a) is straightforward. The introduction of the disease uncertainty increases marginal utility concerning type-
patients but not affect marginal utility concerning type-
patients for the decision make who expresses prudence. Because that the marginal utility decreases with the expenditure of health care. Hence, a prudent social planner shall invest more in health care for type-
individuals than for type-
individuals. In Proposition 1 (b), when the treatment uncertainty is introduced, the partial relative prudence coefficient
causes an increase in marginal utility concerning type-
patients and leaves marginal utility concerning type-
patients unchanged. Thus, the social planner shall invest more in health care for type-
individuals than for type-
individuals.
8More applications for partial relative prudence measure, please see, e.g., [
9,
14,
17–
20].
When the two risks are not independent and are small, by a method that studies precautionary saving in the case of two small risks of Menegatti
[15], we have the following result.
Proposition 2 (a) Suppose that two types of patient only differ in the severity of illness, for which the disease risks are and , respectively. For small risks with , if and , , then the social planner should allocate more resources to type- patients than to type- patients, i.e., .
(b) Suppose that two types of patient only differ in the effectiveness of health care, for which the treatment risks are and , respectively. For small risks with , if , then the social planner should allocate more resources to type- patients than to type- patients, i.e., .
Proof (a) When the disease uncertainty is an additive background risk, by the proof of Proposition 1, it is obvious that . With small risks, from the Taylor approximation (10), we get
where Var and stand for the variance and covariance operators, respectively. It is clear seen that, for small risks with , the inequality holds if and . Owing to (9), we obtain
Hence, the inequalities (16) and (19) imply .
(b) When the treatment uncertainty is a multiplicative background risk, from the proof in Proposition 1, it is realized that . For small risks, by the Taylor approximation (10), we capture
Thereby, for small risks with , the inequality holds if and . By (17) and (19), we obtain that implies .
Given that two small risks are positively correlated. Proposition 2 (a) shows that brings positive direct effect of the dependent disease uncertainty on health care allocation for populations at higher risk and brings positive interaction effect of two dependent risks on it9. Proposition 2 (b) indicates that brings both positive direct effect of the dependent treatment uncertainty and positive interaction effect of two dependent risks. The dependence structure makes the allocation of health care resources rely on the covariance of the two risks and its sign. Therefore, Proposition 2 requires more information about the correlation between two risks and risk preferences than the independent case.
9Notice that the measure of partial relative prudence is usually compared with 2 but not 1, however, Denuit and Rey
[10] demonstrated that the comparison between this measure and its benchmark value 1 is significant for some issues. In addition, our result presents another interpretation for the comparison between the value of partial relative prudence and 1.
Now let us turn to the situation in which the two types of individual differ in the uncertainties on both the severity of the illness and the effectiveness of health care. In what follows, we examine the total influence of two-source uncertainties on health care allocation by comparing the optimal investment in health care of one population in the present of two risks with that of the other population in the absent of them. When the type- individuals face certain disease and treatment, we substitute for in (2). In this case, the optimal allocations to type- and type- patients are denoted as and , respectively.
With two dependent small risks, motivated by the analysis for precautionary saving in two-risk framework of Menegatti
[15], we get the result as follows:
Proposition 3 Suppose that two types of patient differ both in the effectiveness of health care and the severity of illness, for which the treatment and disease risks are and , respectively. For small risks with , if and , , then the social planner should allocate more resources to type- patients than to type- patients, i.e., .
Proof By Lemma 1, it is verified that . When the two risks are small, from the Taylor approximation (10), we have
Hence, for small risks with , the inequality holds if , and . Because (16), (17) and (19), it is evident to view that and imply . The intuition underlying Proposition 3 is as follows. When the two small risks are positively correlated, for the social planner whose preferences fulfill the conditions of Proposition 3, the contemporaneous introduction of the two uncertainties induces an increase in marginal utility involving type- patients without altering marginal utility involving type- patients. As a result, such decision maker will allocate more health care resources to type- individuals than to type- individuals. It should be stressed that there are three separate effects on the expenditure for greater risk population. First, the positive direct effect of the correlated disease uncertainty is controlled by . Second, the positive direct effect of the correlated treatment uncertainty is captured by . Third, the positive interaction effect of two dependent risks is driven by and (or ). It is apparently found that preference conditions in one uncertainty framework are not sufficient to check the joint effect of two types of uncertainty. So, more information on preferences in Proposition 3 is considered.
Finally, instead of two small risks, let us deal with the situation where the two risks possess a stronger correlation. Let
denote the joint distribution of
and
, and
and
denote their marginal distributions, respectively. Lehmann
[21] introduced the following notion to investigate positive dependence.
Definition 1 is positively quadrant dependent, written by , if
The inequality (22) is equivalent to
, which is explained by Lehmann who says "knowledge of
being small increases the probability of
being small"
10. Under the assumption of
risks, adopting the technique that studies precautionary saving of Li
[14], let us re-examine the joint impact of the two-source uncertainties, we obtain the following result:
10Gollier
[22] defined that if for all
,
is non-increasing in
, then
is positive first-degree stochastic dependent (positive
). It could be shown that (see, e.g., [
21])
is a weaker stochastic dependent relationship than positive
.
Proposition 4 Suppose that two types of patient differ both in the effectiveness of health care and the severity of illness, for which the treatment and disease risks are and , respectively. For , if and , , then the social planner should allocate more resources to type- patients than to type- patients, i.e., .
Proof From the proof of Proposition 3, we observe that
. With
, it is captured that
,
and
. By (16), (17) and (19), it is clear that
and
imply
. In comparison with Proposition 3, the preference conditions in Proposition 4 are sufficient to insure the positive effect of
uncertainties on investment in health care of populations at greater risk. Li
[14] illustrated that
is a stronger stochastic dependent relationship than positive correlation. Consequently, under the same preference assumptions, for two small risks, positive correlation is enough to insure more health care investment, but for any two risks, a stronger dependent structure
is needed.
4 The Impact of Inequality Aversion
In the previous section, our analysis is performed in the utility model in which the weights of health benefit for the two types of individual are equal to their shares in population, respectively. Recently, Bleichrodt, et al.
[23] presented a Rank-Dependent QALY model where the weights are assigned to individuals with regard to their health. Under this model, we alter the proportion of patients involved as an equity weighting function that represents the preference towards inequality of the social planner
11. For instance, higher (lower) weight is assigned to the individuals who are worse-off (better-off) under inequality aversion. In the next, we intend to check whether or not the introduction of inequality preference changes the results of health care allocation in Section 3
12.
11The introduction of an equity weighting function allows the decision maker to dissociate attitudes towards health outcome from attitudes towards inequity.
12For more detailed studies of the relationship between health care priority and inequity aversion in other settings, we refer the interesting readers to [
24] and [
25].
It is supposed that type- patients are in a less favorable situation than type- patients. Let , defined over proportion , be the equity weighing function such that to reflect inequality aversion. The social planner selects the investment levels of health care and to maximize the weighted summation of health benefits. The objective problem is represented by the Lagrangian function13:
13Courbage and Rey
[7] focused on the framework with one type of risk, but we consider a more general model of two-source risks.
If the interior solutions exist, denoted by and , then the associated FOCs yield:
In contrast with (3), the optimal investments in health care are relevant to the weighting functions of the two types of individual, and to the degree of inequality aversion. Because that , then (24) becomes
Analogous to Lemma 1, it is easily proved that
14. As in Courbage and Rey
[7], the priority setting results (Propositions 1, 2, 3 and 4) are not affected by the introduction of the equity weighting function in the sense that the decision maker still allocate more health care resources to the patients who are more at risk under the same conditions on the utility function. However, it is not difficult to verify that it augments the difference in the allocation of health care. This is able to be extended to make prediction that the social planner would invest more (less) in health care for populations at higher (lower) risk under inequality aversion than under inequality neutrality.
14The proof is similar to that of Lemma 1 and is omitted.
5 Conclusion
This article provides the preference conditions of the social planner for signing the impact of the contemporaneous presence of disease and treatment uncertainties on health care allocation for higher risk population. Courbage and Rey
[7] have identified the conditions on social preferences under which the
th-degree risk increase in a disease uncertainty or in a treatment uncertainty gives rise to more investment in health care of populations at higher risk. However, when we allow for the simultaneous presence of two uncertainties, the preference conditions in one-risk framework is invalid to guarantee more health care investment. The joint influence of two types of risk takes on ambiguity, which can be eliminated by imposing new restrictions on preferences. We survey different sources of health care priority under uncertainty and find that the dependent structure between two risks plays a critical role in health care decision making. The previous comparative static results are still maintained when inequity aversion is modeled by an equity weighting function. Nevertheless, the inequity preference amplifies the difference in health care allocation. Our conclusions in the paper may be helpful to the exploration of other economic issues in the model of multiple risks.
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