Allocating health expenditures to treatment and prevention

Journal of Health Economics 1 (1982) 265-290. North-Holland Publishing Company

ALLOCATING HEALTH EXPENDITURES TO TREATMENT AND PREVENTION* Dennis R. HEFFLEY University of Connecticut, Storrs, CT 06268, USA Received March 1982, final version recetved August 1982 Rational evaluation of important health sector issues requires more formal analysis of the optimal allocation of resources to treatment and prevention. The suggested approach imbeds a Markov model of the disease process in a simple budget-constrained nonlinear program. Expenditures are allocated to prevention and treatment of a given disease so as to maximize the expected fraction of time spent in the undiseased state by a group of individuals. Randomparameter simulations of a plausible form of the model produce qualitatively different expansion paths for the optimal spending mix, suggesting the need for a flexible policy stance until the parameters of such models can be estimated.

1. Introduction

The marathon health policy debate in the United States has centered on the problem of finding an appropriate mechanism to finance a rapidly increasing flow of health expenditures. To some extent, the preoccupation with funding devices (direct fee-for-service, in-kind transfers, national health insurance, prepaid health maintenance agreements, etc.) may have diverted attention from even more fundamental questions of resource allocation, especially the degree to which health care resources should be devoted to prevention rather than treatment. The treatment/prevention issue enters the health care debate, but often in a superficial or rhetorical way, and again the method of funding usually dominates the discussion. There is tacit recognition of the idea that funding devices may be economically non-neutral - that the introduction of new *This research was jointly supported by the Research Foundation and Summer Faculty Fellowship Program of the University of Connecticut. Much of the initial stimulus for this research came from informal discussions with Llad Phillips (University of California at Santa Barbara); his suggestions during the early stages of the study were particularly useful. Also gratefully acknowledged are the comments of Anthony Boardman (University of British Columbia). Joseph Newhouse (Rand Corporation), Herbert Rauch (Lockheed Research Labs), Donald Shepard (Harvard School of Public Health), and my colleagues at The University of Connecticut: Tryfon Beazoglou, Alpha Chiang, Art Goldsmith, and Stephen Miller. Finally, special thanks are directed to Kathleen Lowney Abdalla for her able research assistance. The usual ‘remaining error’ disclaimers are invoked.

0167-6296/82/ m/$02.75

0 1982 North-Holland

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payment and reimbursement schemes may alter economic incentives and thereby affect the composition of inputs and the production/consumption pattern in the health sector. Proponents of health maintenance organizations, for example, contend that the HMO system is more ‘prevention-oriented’ and less likely to rely upon costly ‘high-technology’ forms of treatment. But, beyond the need to understand how various funding plans influence the treatment/prevention mix of services, there remains the need to examine the more fundamental question of how treatment and prevention contribute to health care objectives within a resource-constrained setting. Before society opts for a funding mechanism or delivery mode that diverts resources from treatment to prevention, or vice versa, it would be helpful to know whether such a shift will actually improve health status and lower costs or merely satisfy particular interest groups. Although there have been a number of attempts to empirically assess the relative impacts of various types of health expenditure,’ there seem to have been few efforts to develop a more formal analytical framework for determining the optimal mix of spending on treatment and prevention, and for describing how this mix should be adjusted as the health budget varies.2 The purpose of this paper is to briefly present a methodology which may lend itself to these issues. Although the model is artificially simple, it does capture many of the essential features of these two forms of health expenditure. The model can be easily expanded and generalized, or it can be imbedded in more conventional choice theoretic models. In addition, the parameters of the basic model have rather natural interpretations. Empirical application of the model is beyond the scope of this preliminary paper,3 so the analysis does not provide any firm policy prescriptions. However, a series of random-parameter simulations of a specific form of the model suggests ‘Auster, Leveson and Sarachek (1969) focus on the relative contributions of medical care and environmental factors to mortality rates, concluding that the environmental factors are more important determinants. Stewart (1971) subdivides health expenditures into four categories (treatment, prevention, information, and research) and attempts to estimate the effect of each type of spending on life expectancy, Meeker (1973), however, has pointed out some specification errors in the latter study. A variety of more precise cost/benefit evaluations have been conducted at the disease-specific level. Cretin (1977), for example, compares cost/benefit ratios for three programs (one preventive and two treatment-oriented) designed to deal with myocardial infarction. *The need for a more general and adaptable analytic device remains, but there have been two notable attempts to tackle this sort of allocation problem at the consumer choice level. Kamien and Schwartz (1973) present an interesting model in which the consumer maximizes expected utility by selecting a level of preventive care. They argue that the resulting decision may be inconsistent with minimum cost provision of health care. More recently, Phelps (1978) analyzes the expected utility-maximizer’s demand for preventive services. While the mode1 itself is not tested, Phelps draws upon the results of a number of empirical studies to illustrate why there may rationally exist a weak demand for preventive services and why insurance coverage often excludes such procedures. %ome of the potential empirical problems are discussed in section 4, however.

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that our policy views about the appropriate mix of treatment and prevention should remain flexible. Optimal strategies may: (i) entail corner solutions’ (e.g., no prevention), (ii) be quite sensitive to the budget level, and (iii) vary across different population groups or different diseases. The treatment/prevention allocation problem can be considered at a variety of decision-making levels. The problem might be broadly formulated in terms of national health expenditure policies (especially for more centralized health care systems), but it’s also easy to conceive of similar decision problems at subnational levels - treatment/prevention choices faced by military health systems, corporate health programs, local public health officials, households, etc. In addition to disaggregating the decison-making unit, it is also possible to consider the treatment/prevention expenditure problem for specific diseases, accidents, or illnesses. In this paper the model is presented in a form that consciously avoids some of the aggregation and distribution problems associated with ‘higher level’ policy decisions. However, it’s a form that seems the most immediately relevant to those engaged in empirical health research and to many ‘frontline’ health service decisionmakers - disease-specific expenditures on treatment and prevention are analyzed for a group of similar individuals (i.e., other influences are assumed to be adequately controlled, either by sample selection or econometric procedures). As an alternative to lumping expenditures for unlike diseases or aggregating dissimilar population groups, more comprehensive health policy models might be constructed by clustering and linking separate models of this sort.

2. The model 2.1. An introduction to Markov models Markov models have been applied to a variety of problems in the health systems field [e.g., Thomas (1968) or Kao (197211.Often these applications are devoid of any explicit form of optimizing behavior and do not provide many economic insights. These ‘non-economic’ models generally serve to describe the probabilistic behavior of a health system or a disease process under a given set of conditions, rather than analyzing how those conditions which govern the system or process might be altered so as to optimize its expected behavior. There are exceptions. Some authors have formally recognized that a health system’s probabilistic behavior is affected by certain input decisions or program. choices, but without further considering how these control variables might be optimally structured [e.g., Bush, Chen and Zaremba (1971) or Ortiz and Parker (1971)]. However, Navarro (1969) has presented a Markov health planning model in which ‘goal seeking’ is

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explicitly analyzed. 4 While Navarro’s model is used to examine the anticipated loads on various sectors of the health care system (primary medical, consultant medical, hospital, nursing home, and domiciliary care), the model presented below focuses directly on the treatment/prevention issue. The following analysis also places a stronger emphasis upon economic interpretation of the optimization process and the resulting health expenditure strategies. A first-order Markov process may be viewed as a sequence of ‘trials’ in which the probability of a particular result for any trial depends at most upon the outcome of the preceding trial. The probability of any sequential combination of results is assumed to be given by a matrix of transition probabilities, I% bij], where pij is the probability that a trial yields outcome j, given that outcome i occurred in the preceding trial. Alternatively, pij can be interpreted as the probability that the process will’move from state i to statej within a specified interval of time. Regularity of the transition matrix P implies the existence of a unique steady-state or limiting probability vector, J such that, regardless of the initial state distribution, the process approaches this limiting distribution as the number of trials becomes sufficiently large.5 Optimization is introduced into the following analysis by allowing transition probabilities to vary in response to certain policy instruments. Once these policy decisions have been enacted and P has been determined, the transition matrix describes the expected state-to-state behavior of the process and uniquely determines the limiting distribution, f, provided that P is regular. This simple modification, linking elements of P to certain control variables, allows one to then ask which policy strategy optimizes some aspect of the system’s equilibrium behavior and how that optimal strategy might be affected by certain parameter changes. 2.2. Assumptions Specification of the model in its simplest and most visible form requires the assumption that for a particular dise%e or illness there exists some binary indicator of health - that in any given period each otherwise similar member of a particular population group can be unambiguously classified as either ‘healthy’ or ‘i11’.6The two states, health and illness, will be denoted by ‘McCall (1970) has employed a similar framework to analyze poverty, and Llad Phillips (University of California at Santa Barbara) has also experimented with the application of this type of model to the criminal justice system. ‘A regular Mar kov process is one characterized by a transition matrix, P, for which there exists some positive integer, k, such that PL contains only positive elements. 6The difficulties of constructing such an index, or even a more continuous measure of health status, are well recognized and have received considerable attention [see Torrance (1976) for a bibliography of some of theses studies and an effor; to integrate the alternative approaches]. Others who have used a binary health status index [Kamien and Schwartz (1973) or Boardman

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subscripts 1 and 2, respectively. For the specific disease, let the probabilistic behavior of each individual in the observed group be given by the common transition matrix,

(1) If pi1 ~(0,l) and p21~(0, l), P is regular and possesses a unique limiting distribution vector, f=(fi f2) =(fi 1-fi), such that fP=f: It follows that f~CP21/~1-PI1+P21),(1-P11)/(1--p,l+p2,)1.

(2)

Rather than regarding the transition matrix as a parameter of the model, it is assumed that elements of P can be altered by changing certain economic control variables. In particular, suppose that pit (the probability that a presently healthy member of the group remains in that state in the next time period) is influenced by the real per capita expenditure per period on preventive care (x), or Pl 1

=m,

g’(x)> 4

g”(x) <

0.

(3)

The range of this function is restricted to the open unit interval by further assuming that g(O)dl>O,

(4)

lim g(x) = 6, (0,l). x-rm

(5)

and

The latter restrictions, together with a similar set of restrictions on the range of pzl, ensure the regularity of P and also serve to introduce some parameters that have very natural and appealing interpretations. Even if nothing is spent on prevention (x=0), it seems likely that there will nevertheless be some minimum probability (0) of remaining free of the disease from one period to the next. There is growing evidence that lifestyle and consumption habits may be important determinants of this parameter and Inman (1977)] would probably agree that it is not the most elegant approach, but it should be pointed out that the longitudinal data normally required to implement the following type of model are often dichotomous. If data permit, the binary description of health status can be modified by including additional states (see section 4). The two-state assumption is somewhat less restrictive in the present disease-specific context, where the presence of the disorder may be marked by fairly clear symptoms, than in more highly aggregated models where ‘general health’ is being measured.

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for certain diseases, but 8 will also reflect certain natural body defenses that are largely unaffected by private or collective human decisions. The present model assumes a group of similar individuals; in a less homogeneous population 8 might differ across individuals. As will be noted later, variations in such health parameters, as well as economic ones, may help to explain observed differences in individual health spending decisions. The upper bound parameter (8) reflects the notion that even if extremely large per capita sums are spent on prevention, there will always remain some chance (at least 1-e) of a healthy individual contracting the disease. Both (4) and (5) are concessions to reality rather than severe restrictions on the generality of the model. The other control variable is y, the real per capita expenditure per period on treatment of the specific disease. The rate of therapeutic expenditure is assumed to influence the probability (p& that a presently ill individual will be restored to health in the next period, or P2r

=

MY),

MY)> 0,

h”(y)< 0.

(6)

In addition, h(O)=p>O,

(7)

and lim hQ=j.k(p, Y-+@

1).

(8)

The interpretations of ~1and fi are analogous to the interpretations of 8 and & The positive intercept (p) reflects the presence of certain natural curative mechanisms (e.g., antibodies) which operate even when no resources are channeled into treatment. The upper bound (fi) on pzl implies that there will always remain some positive probability of non-recovery (pz2) between time periods (i.e., p22> 1 -fi> 0), even if arbitrarily large amounts are spent on treatment. Note that (3) and (6) may be viewed as rather crude disease-specific ‘production’ functions for the preventive and curative activities of the health sector. For simplicity, both functions are assumed to exhibit positive, but strictly decreasing, marginal returns. For the analysis that follows, it is convenient to define the respective inverse functions,

(9) and (10)

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The model circumvents the usual problems of defining acceptable proxies for health sector outputs by treating pll and pzI as the relevant products.’ This procedure has some intuitive appeal and may be a useful way of coping with the problem of intangible outputs - one of the most serious and commonly cited impediments to the economic analysis of service industries. Now consider the following allocation problem. Given a particular amount (S real dollars per capita) available in each period for total expenditures on the disease, how should funds by apportioned to prevention (x) and treatment (j$? Response to such a question clearly requires some decision criterion. Within the context of the simple Markov model described above, a reasonable objective is the maximization of fr, the expected fraction of time the observed group spends in the disease-free state. Note that since g’(x)>0 and

vllh1 =Pzlm -PI* +P21)2>o,

(11)

any increase in spending on preventive care will contribute to the objective. But, since h’(y)>0 and

any dollar of the budget diverted from treatment to increase prevention will also tend to reduce fr as pzl falls. The problem is obviously one of finding the appropriate balance between the two types of outlays, where the marginal contribution to fi of preventive spending equals the marginal contribution of expenditures on treatment to fr. Formal optimization techniques support this fundamental economic reasoning, but before proceeding in that direction, a graphical exposition of the model will prove useful and also help to reveal the appropriate method of optimization. 2.3. Graphical analysis Three key elements of the model have been introduced: (i) g(x), a functional relationship between expenditures on preventive care and the probability that a healthy individual remains in that condition, (ii) h(y), a functional relationship between expenditures on treatment and the probability that an ill person is returned to health, and (iii) a budget constraint on total outlays per period for the particular disease. The latter may be written as B-x-y=O,

(13)

‘This procedure is also compatible with Grossman’s (1972) notion that individuals demand ‘good health’ rather than health services per se.

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D.R. Hefley, Allocating health expenditures

These three components of the model, taken together, fully determine the probabilistic alternatives, a set of feasible combinations of pI1 and p21. These alternatives are described by the expression obtained by using (9) and (10) in (13), (14) The (pI1,pzl) combinations that satisfy (14) form a transformation surface that describes the health system’s capacity to substitute pll for p21, given the per capita financial resources per period (B) earmarked for the disease. The following four-quadrant diagram (fig. 1) shows how this transformation surface is derived from g(x), h(y), and the budget constraint. Any expenditure combination (x, y) along the budget constraint in the lower right quadrant implies, via g(x) and h(y), a particular feasible combination of transition probabilities (i.e., an attainable transition matrix, P). The concave transformation surface, RS, in the upper left quadrant is simply the set of all probability pairs associated with expenditure

PII

Fig. 1. Derivation of the transformation surface(RS).

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combinations that exhaust the budget. While any of the combinations in the shaded region below the surface are also feasible, the nature of the objective function (discussed below) ensures that a maximum value of fi will always be obtained somewhere on the outer boundary of the set. Again, the problem is to select some feasible expenditure mix, and therefore some combination of transition probabilities, which maximizes ,fi, the expected fraction of time the group spends in the healthy state. Recall from (2) that fi, the first element off, is a function of both prl and p2!,. The form of this objective function becomes clearer if the expression for fi is rewritten as p11= 1+ Cl -k!fl)lP2l~

All combinations along a negatively sloped ray from (but not including) the point (pl 1,p21)=(l,O) will yield a particular value of f,. For larger values of the fraction fI, the absolute value of the slope coefficient, [l -(l/jI)], is smaller. Hence, the less steeply sloped the contour ray, the larger the value of fi associated with the various probability pairs along that ray. The objective

Fig. 2. Determination of the optimal transition structure (pr,,p!,j and

spending mix (x*+Y’).

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is to attain a position on the highest ray (i.e., maximize Jr), subject to the transformation constraint, as illustrated in fig. 2.8 The determination of an optimal transition structure (pf,,pfr) implies an optimal spending mix (x*, y*). The optimum will typically occur at the point of tangency between the transformation surface and a contour ray of the objective function (as in fig. 2), but the possibility of a corner solution should not be excluded. To allow for both interior (tangency) and corner solutions; the problem is best formulated as a simple nonlinear program. 2.4. Mathematical analysis In nonlinear programming format, the health expenditure allocation problem can be stated as (16) subject to the transformation constraint,

~-g-‘(p~Wf-‘~,,)~Q

(17)

and the lower bound restrictions on the choice variables, Pll

-WA

(18)

PZl

-Im

(19

and

The asymptotic nature of properties (5) and (8) make it unnecessary to incorporate upper bound constraints in the formal program, since such constraints would never be binding for finite health budgets. Expressed in Lagrangean form, the problem is to

‘The problem can be graphically depicted in an alternative, but equivalent, way. Since any contour ray of the objective function represents a particular value of $,, egch point along such a ray could be mapped [via g(x) and /r(y)] into (x,y)-space in the lower right quadrant. The resulting locus connects all spending combinations which yield that particular value of fr, and might be described as an ‘iso-health’ contour. There exists one such locus for each value of fr c[O, 11, with contours further from the origin representing larger values of f,. Given the conventional properties of g(x) and h(y), these loci will be convex to the origin. The problem of maximizing f, subject to the budget constraint is solved by locating the expenditure combination on the budget line which rests on the iso-health contour furthest from the origin.

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Three distinct types of solutions require examination. Corner solutions may occur at R or S along the transformation surface (see fig. 2); the third and probably most common solution type is an interior optimum at some point on the transformation surface between the endpoints R and S. The Kuhn-Tucker (1951) conditions can be used to derive a general optimality requirement that covers all three types of solutions,

At an interior optimum, where neither of the lower bound restrictions hold (i.e., pr 1 > 0 and p21>p), both A2 and A3 vanish and (21) provides the usual marginal conditions. For an R-type (all-prevention) corner solution, where PG~ and P21 =p, A, remains zero but A310, implying that the marginal contribution to fi of the Bth dollar of prevention [(8fi/8p1i)(dpil/dx)] exceeds the marginal contribution of the first dollar of treatment [($J8p2i)(dp2,/dy)] by an amount equal to AJ(dp2,/dy)20. Similarly, an alltreatment strategy (or S-type solution, where p1 1 = 0 and pzr > p) is optimal only if A22 0, A3= 0, and the marginal contribution of the final dollar spent on treatment exceeds the marginal contribution of the first dollar of prevention by 12(dp, ,/dx) 2 0. These conditions suggest a practical algorithm for locating the optimal solution in this simple form of the model?

6) At R [i.e., where

p1 1 =g(B) and pzl = p], evaluate and compare LHS =((afi/8p11)(dpl,/dx) and RHSr(af,/ap21Kdp21/dy). If LHS 2 RHS, stop; otherwise continue. (ii) At S [i.e., where p1 l = 0 and p21=h(B)], evaluate and compare the same expressions. If LHS $ RHS, stop; otherwise continue. (iii) Solve for the interior point (p,, ,P~~), where LHS= RHS and B-g-‘(p,,)-h-‘(p,,)=O.

This algorithm will be used in the simulations that follow (section 3), but first it’s worth noting the practical applications of certain shadow-price information derived from the formal nonlinear program. Consider a case in which the entire available budget is optimally spent on treatment (S-type solution). Since x=0, pll = 8 and, hence, rZ220. Under such conditions, an increment in 8 (the ‘natural’ or base probability of remaining healthy) will permit an increase in fi, as illustrated in fig. 3. Evaluation of A2 ‘The algorithm relies upon sufficiency of tt;e Kuhn-Tucker (K-T) conditions. Although the K-T conditions are often only necessary, Arrow and Enthoven (1961) have provided a set of conditions which, if satisfied, ensure that the K-T conditions are also suflicient. The present problem can be shown to comply with the Arrow-Enthoven quasiconcavity conditions for the objective function and each of the constraints, and at least one of their four subsidiary conditions is met.

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1.0

Fig. 3. EtTectof increasing the base probability of remaining healthy (0).

at this S-type corner solution provides a measure of the marginal benefits (expressed as a change in the fraction of time the group spends in the disease-free state) of increasing 8. Recall that 8 may be affected by consumption habits, lifestyle, etc. A comparison of the marginal costs of altering such habits (through education, taxes/subsidies, insurance terms, etc.) with the marginal benefits (A,) of increasing 0 might further contribute to rational decision-making in the health sector. Analogous information about the marginal benefits of raising p is provided by &, while A1 measures the marginal increase in J1 as the health budget (B) expands. Estimates of AI for various disease-specific allocation problems would be particularly useful in establishing individual budget levels for competing disease-control programs. However, such applications may require formal recognition of interdependencies between various illnesses and diseases. 3. Simulations Reasonable restrictions have been placed upon (3) and (6), but no specific functional forms have been suggested for simulation (or future estimation) purposes. The following bounded exponential functions possess all of the desired properties listed in (3)-(5) and (6)-(S), respectively, p1,

q(x)=

&-(8- e)e-,

Pzt =~(Y)=P-w-P)e-~y,

e&(0,1),

e
COO,

p*jk(O, l), jlLo. *

(22) (23)

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The parameters of these forms are readily interpreted. The lower bounds on Pll and pzl, respectively, are 9 and p; while 6 and fi are the asymptotic upperbounds on the same variables. Furthermore, since a=(dp,,/dx)/ @-pl I), and parameter a is the rate of increase in p11 due to an increase in X, relative to the potential improvement in p 11. The parameter /? is analogously interpreted. If the parameters (0, & a; p, ii, /?) of these functional relationships are known for a particular disease and a specific population group, then for any per capita budget (B) one can solve for the optimal combination of transition probabilities and the associated mix of spending on prevention (x*) and treatment (y*).lo Moreover, as the parameter B is varied, an expansion path for the optimal spending mix can be generated, as in fig. 4. As the budget increases from B, to B2, the transformation surface expands outward from R,S, to R2S2, and the optimal mix of prevention and treatment changes

Fig. 4. Derivation of the expansion path (EP) for spending on prevention (x) and treatment (y).

‘OFor details, see the appendix.

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from (xf, yt) to (x2*, yf). Allowing B to vary continuously, an optimal expansion path (similar to EP in the above figure) can be mapped. The purpose here is not to undertake an extensive sensitivity analysis, but rather to demonstrate the dependence of the optimal mix on the size of the health budget, and to illustrate the qualitatively different types of spending strategies that may be optimal under alternative sets of parameters. The parameters for the first two simulations were randomly generated, subject ;~nly to the a priori restrictions that 0~6 and p
0= 0.53, 6= 0.97, 01=0.30; /I = 0.29, fi = 0.65.

/I = 0.40.

B

P:,

PZl

X*

Y*

ff

0.0 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0

0.53 0.53 0.53 0.53 0.54 0.56 0.67 0.80 0.87 0.91 0.93

0.29 0.32 0.34 0.37 0.38 0.38 0.39 0.40 0.41 0.43 0.46

0.00 0.00 0.00 0.00 0.06 0.26 1.22 3.12 4.96 6.72 8.36

0.00 0.20 0.40 0.60 0.74 0.74 0.78 0.88 1.04 1.28 1.64

0.38 0.40 0.42 0.44 0.45 0.47 0.54 0.66 0.76 0.83 0.88

Simulation II:

B 0.0

0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0

0=0.06,

$=0.14,

a=0.22;

j~O.16,

fi=O.68,

B=O.68.

P:,

Pfl

X*

Y*

f:

0.06 0.06 0.06 0.06 0.06

0.16 0.23 0.28 0.33 0.38 0.42 0.55 0.65 0.66 0.67 0.67

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.83 2.37 3.46

0.00 0.20 0.40 0.60 0.80

0.15 0.19 0.23 0.26 0.29 0.31 0.37 0.41 0.42 0.42 0.43

0.06 0.07 0.09 0.10

;: 4.00 5.17 5.63 6.54

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Fig. 5 illustrates the expansion path for each of the two sets of parameters. Note that in both cases optimality requires that smaller health budgets be allocated entirely to treatment. In the first . simulation, spending on prevention is inefkient until B=0.73. In simulation II, prevention does not enter into the optimal mix until B=4.93. These vertical segments of the expansion paths are generated by a sequence of S-type corner solutions. Recall that such a solution is optimal when the marginal contribution to f, of the Bth unit of spending on treatment equals or exceeds the marginal contribution of the first unit of spending on prevention. It is therefore tempting to attribute this initial dominance of treatment to the fact that /I > M in both of the above simulations. Unfortunately, it is not so simple, as demonstrated by the next two simulations. Simulations III and IV are ‘mirror images’ of simulations I and II, respectively. In each case, the parameters of the prevention and treatment functions, g(x) and h(y), have been interchanged. For simulation III, (&6; cs;CL, p, /3)= (0.29,0.65,0.40,0.53,0.97,0.30), and for simulation IV, (~,~~~;~,p,~)=(0.16,0.68,0.68;0.06,0.14,0.22), so for both cases we now have cr>/3. In simulation IV, this swapping of parameters leads to an allprevention strategy (x* =B,y*=O) at lower health budgets (Bs 1.12) - just the opposite of the pattern in simulation II. However, in the third simulation we also have or>& yet for lower health budgets (Bs0.39), fi is maximized

Fig. 5. Expansion paths for simulations I and II.

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Allocating health expenditures

by spending all available funds on treatment, as in simulation I where P>cc. The expansion paths for simulations III and IV are shown in fig. 6.’ ’ While # and /3 are important parameters, any comparison of the marginal impact of the two types of health spending entails all of the functional parameters. Using (1l), (12) and derivatives of (22) and (23), the marginal contributions of preventive and therapeutic expenditures are given by Afi

afi dp,,

dx=Z&ZF=[l

a(8-8)@-(/i-p)e-~y]

(24)

-B+(~-~)e-‘“+~-~-cl)e-~y]ze~y

and Afi

afl dp,,

dy=ap21dy=[I

fl(fi-~)[1-~+(8--e)e-“3

(25)

-8+(g_e)e-ax+P-~-~~-~Y]2eSY’

Not only is the marginal impact of each type of spending dependent upon all six of the parameters, but also upon thg Tisting level of spending in each of the two categories (i.e., x and y). Even this structurally simple two-state model with well-behaved functional forms points to a rather complex interdependence between treatment and prevention.

IO

6

1.12 2

4

6

0

IO

Fig. 6. Expansion paths for simulations III and IV. “Tables for simulations III and IV are available from the author on request.

w

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4. Qualitative implications and possible extensions The model developed in section 2 is far too simple to provide any definitive answers about the process of optimally allocating health sector resources to treatment and prevention. The simulations presented in section 3 are based upon a bounded exponential form of the model - a reasonable and fairly flexible form, but still restrictive. The analysis of the model and results of the sample simulations, though, do suggest several things about possible optimal spending patterns. First, it may be fully optimal for the entire amount budgeted for a particular disease to be spent on either treatment or prevention, especially at low budget levels. This sort of ‘plunging’ behavior may worry Llose with strong marginalist instincts, but corner solutions have equally plausible and consistent economic interpretations.12 Some are libely to find the possibility of an all-treatment solution particularly troubling, for there is a tendency to regard prevention as being inherently more rational than treatment. But, while spending on prevention may lower the probability of becoming ill (i.e., raise pll), it also reduces the amount available for treatment if and when the illness dots occur. The bias toward preventive strategiee is particularly evident in the political sphere. For example, without presenting sound empirical evidence about the relative costs or effectiveness of treatment and prevention, Magnuson and Segal (1974, p. 180) assert that: ‘It is clear that the most economical and humane method of dealing with illness is to prevent it.’ But, as Cretin (1977, p. 174) aptly notes: ‘Advocates of measures to prevent or postpone the onset of illness have not always realized that a good treatment may be preferable to a mediocre preventive program.’ It may be both inhumane and inefficient to subject a par,ticular group to a preventive screening procedure characterized by a high rate of false positives13 or an innoculation program with potentially debilitating side-effects, and then fail to provide adequate therapy to diseased individuals because resources have been exhausted on questionable preventive measures. In recent years there have appeared a number of cost/benefit studies of alternative strategies for dealing with specific diseases. Some of these studies seem to indicate a role for mixed strategies. Cretin (1977), for example, calculates roughly equivalent cost/benefit ratios for preventive, ambulance, and in-hospital methods of dealing with myocardial infarction. But other studies [Moulding (1971) or Stason and Weinstein (197711 find that, for certain diseases and/or population groups, preventive measures do not represent a very rational investment of resources. At a less disease-specific lZThe theoreticai possibility of zero-prevention corner solutions has also been noted by Phelps (1978, pp. 186-187). 13For an excellent discussion of this problem, see Phelps (1978, pp. 195-202). C

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level, Stewart (1971) presents evidence which suggests that an’ all-prevention strategy might be optimal for the developing country with a small health budget. I4 Obviously there already exists a broad range of empirical findings concerning the appropriate mix of treatment and prevention. The present analysis cannot corroborate, consolidate, or reconcile any of these previous empirical results, but is does suggest that the diversity of findings may reflect differences in the parameter: underlying each study rather than severe inconsistencies or errors of analysis. The richness of possible solutions to the treatment/prevention allocation problem also suggests that the most effective way to approach the issue may be from a highly disaggregated diseasespecific Ieve:, but researchers seem more inclined to follow this path than do policy-makers. A second implication of the model involves the dependence of the optimal spending mix on the budget level. The optimal mix of expenditures is ‘:kely to change as the health budget varies, and this pattern of change (i.e., properties of the expansion path) may differ substantially for alternative diseases or various population groups. Fig. 7 presents the relationship between the budget level (II) and the optimal ratio of preventive-to-total expenditures (x*/B) for each of the four simulations in section 3.

The qualitatively different spending patterns generated by varying parameters of the model suggest that it may be best to view with caution any recommendations that one group adopt the spending mix of another group whose budget, personal characteristics (age, sex, lifestyle, health history, etc.), or access to medical technology is measurably different. Similarly, observed differences in the proportion of the budget spent on prevention should not automatically be interpreted as evidence of suboptimal behavior by one group or another. For example, a low-income family’s revealed preference for treatment over prevention may represent efficient decision-making within a tightly constrained budget and a particular parameter environment, rather than myopia or a lack of knowledge about the relative benefits of treatment and prevention. Such issues require further empirical research, but in the absence of such information we should perhaps refrain from lamenting the patterns of expenditure on various illness or diseases registered by certain groups. A third feature of the simulations may also be of interest. Another glimpse at fig. 7 reveals that the optimal ratio of prevention to total outlays for a particular ailment need not be a monotonic function of the budget level. Broad and unsubstantiated claims that a larger fraction of resources should be devoted to prevention as the health budget increases should be viewed with skepticism. Even in cases where this prescription seems initially “Meeker (1973) has challenged Stewart’s econometric procedures and economic interpretations, casting some doubt upon the conclusion that treatment is ineffective and the training of physicians a bad investment.

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Fig. 7. The optimal ratio of preventive-to-total spending (x*/B) for alternative budget levels; simulations I-IV.

warranted (simulations I-III), the optimal prevention-to-budget ratio may peak and then decline as B becomes sufficiently large (simulation III). In other cases (simulation IV), such a recommendation may be even more misleading and costly. Finally, although we can derive an optimal expansion path for a given problem, it is not clear precisely how the mixture of prevention and treatment should be adjusted as the functional parameters of the system change. Suppose, for example, that some breakthrough in prevention causes ct to suddenly increase. The new set of parameters implies a new transformation surface and will generally call for a new optimal mix of treatment and prevention, but the path this adjustment process should follow is not determined within the present stochastic equilibrium model (see fig. 8a). In a similar vein, the model says virtually nothing about which path to the optimal point should be followed by a system that is presently operating at some interior point of the feasible set (fig. 8b). A more dynamic specification of the model is required to answer questions of this sort.15 ‘)Herb Rauch (Lockheed Research Labs, Palo Alto) has suggested an interesting reformulation of the model as a problem of optimal control. Not only the present problem, but also more complex treatment/prevention problems involving intertemporal allocation across population cohorts seem to lend themselves to control theory techniques. As in many such applications, though, modeling the problem may be less dillicuh than developing efficient computational algorithms.

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a

b

1.0

Fig. 8. Dynamic adjustment (a) from one optimum to another, and (b) from suboptimal point to an optimum.

The preceding discussion of qualitative implications suggests at least one way in which the simple model might be extended. But there are some other appealing non-dynamic modifications that would help to provide a more comprehensive and sophisticated analysis of the health expenditure allocation problem. Even for a particular disease, health is not a binary phenomenon but rather a continuum of states. This continuum can be more closely approximated by including additional states of health. An obvious omission in the two-state model is the state of death. Non-trivial sums are presently allocated to preventing the ill from dying, protecting the healthy from sudden death (especially if one includes expenditures on such things as automobile safety equipment, highway design, etc.), and even on restoring the seemingly ‘dead’ to some less terminal state. The magnitude of these expenditures, and the ease with which they could be fitted into the format of the model, make death a logical choice for an additional state. But the nature of this special state raises some methodological problems. Despite the existence of spectacular resuscitation procedures, death is essentially an absorbing state - a state from which the probability of transition into any other (terrestrial) state is zero. Inclusion of such a state in the present model would make the limiting distribution fairly uninteresting. Since one element of this stationary distribution (the fraction of time spent by the group in the state of health) serves as the maximand in the allocation problem, the addition of an absorbing state is more than a technical nuisance. Fortunately, there is an appealing way out of this apparent bind. In an absorbing Markov chain, absorption occurs with probability one.16 It is common under these circumstances to nevertheless ask: (i) how long the process will take to reach *he L1Lrabsorbing state from any other given state, and (ii) how many times the process is expected to visit any given state before ‘%ee Kemeny, Mirkil, Sell, and Thompson (1959, p. 404).

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absorption occurs. The answers to these and a host of related questions can be obtained quite easily.” Since, in an absorbing chain, the focus shifts from the long-run distribution to the span and pattern of the process, the objective function must be reformulated accordingly. But, if anything, the set of possible objectives is even richer. For any absorbing transition matrix, one can calculate the expected number of periods required to reach death from the healthy state. Thus, if elements of this absorbing transition structure are linked to various forms of expenditure or ways of dealing with the disease in question, one might ask which spending mix maximizes life expectancy. Such an objective reflects a concern for the duration of life but not necessarily the quality of life (e.g., a strategy might maximize life expectancy by simply prolonging the lives of patients who enter a comatose state). The latter might be formally incorporated as an objective, however, since it is also possible to calculate for any absorbing transition matrix the mean number of periods an individual occupies a given state (e.g., healthy) before dying. It is even possible to develop an objective which places some weight on the certainty of health, since variances for these means (i.e., the mean occurrence of each nonabsorbing state) can also be calculated for any absorbing transition structure. Expansion of the m.odel by including these additional states clearly complicates matters, but it also makes the model more worthy of the effort that would be necessary for any serious empirical implementation. Another extension of the basic model might better approximate the reality of health sector decision-making. The model presented in this paper is a long-run stochastic equilibrium model, where the budget level and the technologies of prevention and treatment are the only effective constraints on the choice of a spending mix. In effect, the decision-maker is permitted to ‘start from scratch’ and choose an optimal strategy from among the full set of feasible alternatives. This may be a luxury that health policy-makers seldom enjoy. In a highly institutionalized health sector with disparate interest groups, the allocation of health expenditures may bs subject to additional constraints. It may be useful to briefly consider the effects of one such set of constraints. Suppose that there exists an initial allocation of expenditures to the prevention and treatment of a particular disease (x,,yi), which may or may not be optimal. Additional funds are to be made available for expenditure on this disease, and the question is: Wow should these incremental funds be allocated? Decision-makers will not be permitted to reduce present expenditure flows on either prevention or treatment, perhaps due to political constraints or institutional rigidities; they may only increase one or both forms of expenditure such that X~&Q and y2sy1. What does the present model have to say about such a decision? “See Chapter III of Kemeny and Snell(l959).

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First, note that this long-run stochastic equilibrium model has very little to say if the incremental expenditure is simply a one-time, transitory increase in out-lays. Only sustained increases in the annual budget (B) will permanently influence the transition structure (P) and, hence, the limiting distribution (f) over alternative states of health. If the incremental funds do represent an increase in the annual commitment of resources to the problem (A@, then the model is applicable, but it must be modified to reflect the ‘status quo’ constraints discussed above. Fig. 9 illustrates two cases.

b

Fig. 9. Non-binding (a) and binding (b) ‘status quo’ constraints.

In fig. 9a, the initial budget has been allocated to achieve a particular combination of transition probabilities (point 2,); in this example the initial allocation is optimal, but it need not be so for the analysis to hold. Suppose that a new expanded budget is available for expenditure on this disease, so that the feasible set of probability combinations shifts outward. How should this new budget be allocated, given that expenditures on neither prevention nor treatment may be reduced? If there are no changes in the functional relationships, g(x) and h(y), between expenditure levels and the transition probabilities, the ‘status quo’ constraints on x and y imply that only a move to the northwest of Zi is permissable. If, as in fig. 9a, the new long-run optimum (2,) lies to the northwest of the initial combination, then both of the ‘status quo’ constraints are non-binding and the prescriptions of the model are perfectly applicable. Fig. 9b, though, illustrates a case where one of the added constraints is binding. Full optimality would require an allocation of expenditures associated with point Zz, but this would entail a lower value of p2i and, hence, a reduction in spending on treatment. Under the added constraints, the second-best choice is Zl,, where all of the budget increment is used to increase pI1 by raising expenditures on prevention. The move from Z1 to Z;

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clearly increases the expected fraction of time the population will spend in the healthy state, but since Z2 lies on a lower $,-contour ray that does Z2, the cost of adhering to the binding ‘status quo’ constraint is the further increase in f1 that is foregone in order to avoid reducing expenditures on treatment. It should be pointed out that one can construct a completely analogous example in which the other constraint is binding and imposes a similar health cost on the population in question. If the mathematical form of the model were respecified to formally recognize these ‘status quo’ constraints, computed values of the associated Lagrange multipliers would provide direct information about the size of these costs. In addition to formally incorporating supplemental constraints, one might further improve the applicability of the model by allowing for higher-order Markov processes. The present model assumes a simple first-order process, in which the probability of entering a particular health state is dependent upon one’s health status in the preceding period only. An nth-order Markov process would consider the individual’s sequential health status’in each of the preceding n periods. This extension allows the individual’s health history to influence his or her response to preventive or therapeutic intervention. Such an extension may be particularly important in dealing with the sort of mixed risk populations analyzed by Shepard and Zeckhauser (1980). There are undoubtedly a variety of other extensions (reformulation as a semi-Markov model, aggregation of submodels for various income classes, age/sex groups, etc.) which might be fruitful, but it seems appropriate to conclude with some mention of the major empirical problems associated with a model of the sort presented in this paper. The estimation of parameters for even the two-state model is potentially difficult for two principal reasons. First, the estimation of transition probabilities generally requires longitudinal microdata. The sequence of changes in health status, not merely the number of changes between two points in time, is important in determining the transition probabilities. While Lee, Judge and Zellner (1970) have devised some imaginative econometric procedures for estimating transition matrices from aggregate time series data, their methods do not solve the second basic empirical problem, Namely, it is not sufficient to simply estimate P as a matrix of parameters; the optimization structure of the model requires that changes in elements of P be related to changes in various types of health expenditures. In short, since the parameters of g(x) and h(y) must be estimated, the longitudinal data must include information on amounts and types of health expenditures for individuals as well as their health status. Controlled longitudinal studies of specific diseases may be the cleanest source of experimental data for these estimations. Sources of adequate retrospective data are more difficult to imagine. The most suitable testbed for such a model might be a closed military health system, in which individual health status and services received are regularly, if imperfectly, recorded (e.g., sick

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bay reports). HMOs may also serve as useful data soucses if one can control for ‘outside’ preventive and therapeutic services received by HMO members, These problems are not insurmountable, but they do place empirical implementation beyond the resource constraints of this preliminary study.

Appendix

Applying the algorithm outlined in section 2 to the suggested functional forms, (22) and (23), it can be shown that a corner solution will occur at R (i.e., LHS 2 RHS), where pfI =g(B)=$-(Q-6)e-aB, X+=B,

PC = I4

y*=o,

if the parameters of the problem satisfy the condition

(A4 Similarly, a solution will occur at S (i.e., LHS 5 RHS), where Pf I = e, X*

=o,

pgr =h(B)=fi-(ji-p)e-@, y*=B,

if the parameters are such that

(A-2) If neither of the above conditions hold, there exists an interior optimum, found by iterativeiy solving for the root of

wx~--Pl 1)-(~)(-~~C~(~-~11)+8(1-~11)1=0,

(A.31

and then substituting this solution value (pf,) into the following expression to obtain paI, p2 1 = P - Kfi - de +‘I LW- @A@- P 11HP’“.

(A-4)

D.R. Hefley,

Allocating health expditures

Optimal spending levels (x* and y*) are finally obtained by using in the inverse forms of (22) and (23, namely,

289 pfl

and

pfl

and

References Arrow, Kenneth and Alain Enthovefi, 1961, Quasi-concave programming, Econometrica 29, 779-800. Auster, Richard, Irving Leveson anlij Deborah Sarachek, 1969, The production of health: An exploratory study, Journal of Human Resources 4.41 l-436. Boardman, Anthony and Robert Inman, 1977, Early life environments and adult health, Paper presented at session sponsored by the Conference on Social Sciences in Health, American Public Health Association Meetings (Washington, DC). Bush, James, Milton Chen and Joseph Zaremba, 1971, Estimating health program outcomes using a Markov equilibrium analysis of disease development, American Journal of Public Health 61, 2363-2375. Cretin, Shan, 1977, Cost/benefit analysis of treatment and prevention of myocslrdial infarction, Health Services Research 12, 174-189. Grossman, Michael, 1972, On the concept of health capital and the demand for health, Journal of Political Economy 80, 223-255. Kamien, Morton and Nancy Schwartz, 1973, Payment plans and the eficient delivery of health services, Journal of Risk and Insurance 40,427-436. Kao, Edward, 1972, A semi-Markov model to predict recovery progre!‘s of coronary patients, Health Services Research 7, 191-208. Kemeny, John and J. Laurie Snell, 1959, Finite Markov chains (Van Nostrand, Princeton, NJ). Kemeny, John, Hazleton Mirkil, J. Laurie Snell and Gerald Thompson, 1959, Finite mathematical structures (Prentice-Hall, Englewood Cliffs, NJ). Kuhn, Harold, and Albert Tucker, 1951, Nonlinear programming, in: J. Neyman, ed., Proceedings of the second Berkeley symposium on mathematical statistics and probability (University of California Press, Berkeley, CA) 481-492. Lee, Tsoung-Chao, George Judge and Arnold Zellner, 1970, Estimating the parameters of the Markov probability model from aggregate time series data (North-Holland, Amsterdam). Magnuson, Warren and Elliot Segal, 1974, How much for health? (Robert B. Lute, Washington, DC). McCall, John, 1970, An analysis of poverty: A suggested methodology, Journal of Business 43, 31-43. Meeker, Edward, 1973, Allocation of resources to health. revisited, Journal of Human Resources 8,257-259. Moulding, Thomas, 1971, Chemoprophylaxis of tuberculosis: When is the beelefit worth the risk?, Annals of Internal Medicine 74, 761-770. Navarro, Vicente, 1969, A systems approach to health planning, Health Services Research 4, !N111. Or&, Jorge and Rodger Parker, 1971, A birth-life-death model for planning and evaluation of health services programs, Health Services Research 6, 120-143. PhelDs. Charles, 1978. Illness prevention and medical insurance, Journal of Human Resources 12,suppl., 183-20;. Shepard, Donald and Richard Zeckhauser, 1980, Long term effects of interventions to improve survival in mixed populations, Journal of Chronic Diseases 33,413-433.

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Stason, William and Milton Weinstein, 1977, Allocation of resources to manage hypertension, New England Journal of Medicine 296,732-739. Stewart, Charles, Jr., 1971, Allocation of resources to health, Journal of Human Resources 6, 103422. Thomas, Warren, 1968, A model for predicting recovery progress of coronary patients, Health Services Research 3,185-213. Torrance, George, 1976, Health status index models: A unified mathematical view, Management science 22,990-1001.