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Cost-effectiveness analysis and capital costs
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Soc. Sci. Med. Vol. 46, No. 9, pp. 1183±1191, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0277-9536(97)10046-6 0277-9536/98 $19.00 + 0.00
COST-EFFECTIVENESS ANALYSIS AND CAPITAL COSTS GOÈRAN KARLSSON* and MAGNUS JOHANNESSON Stockholm School of Economics, P.O. Box 6501, S-113 83 Stockholm, Sweden AbstractÐTraditionally, economic evaluations in terms of cost-eectiveness analysis are based, explicitly or implicitly, on the assumption of constant returns to scale. This assumption has been criticized in the literature and the role of cost-eectiveness as a tool for decision making has been questioned. In this paper we analyze the case of increasing returns to scale due to ®xed capital costs. Cost-eectiveness analysis is regarded as a tool for estimating a cost function. To this cost function two types of decision rules can be applied, the budget approach and the constant price approach. It is shown that in the presence of ®xed capital costs the application of these two decision rules to a speci®c patient group will give dierent results. The budget approach may lead to suboptimizations, while using the price as a decision rule will give optimal solutions. With ®xed capital costs and when an investment can be used for treating several patient groups, these groups are no longer independent. Therefore the cost-eectiveness analysis has to be performed simultaneously for all patient groups that are potential users of the investment. # 1998 Elsevier Science Ltd. All rights reserved Key wordsÐcost-eectiveness analysis, decision rules, increasing returns to scale, capital cost, indivisibilities, economic evaluation
INTRODUCTION
Traditionally, economic evaluations in terms of cost-eectiveness analysis are based, explicitly or implicitly, on the assumption of constant returns to scale. This assumption means that the cost per treated patient and the eectiveness per treated patient are independent of the size of the program. The decision rules of cost-eectiveness analysis have previously been stated in the literature (Weinstein and Zeckhauser, 1973; Weinstein, 1990; Johannesson and Weinsten, 1993; Karlsson and Johannesson, 1996). For many treatment programs the assumption of constant returns to scale can be justi®ed. However, there are programs that exhibit high investment costs, for example treatment with the gamma knife and radiation therapy. In these cases the capital cost for the equipment makes up a signi®cant share of the total cost, and at least up to a certain level the average cost per treated patient is decreasing with the size of the program. Therefore the assumption of constant returns to scale does not hold. There is no consensus in the literature as to whether or not the assumption of constant returns to scale is an acceptable approximation to the real world situation. The existence of non-constant returns to scale, among other concerns, has led Birch and Gafni to the conclusion that cost-eectiveness analysis is not a practical tool for decisions about resource allocation (Birch and Gafni, 1992; Birch and Gafni, 1993). Johannesson and Weinsten *Author for correspondence.
(1993), on the other hand, argue that the traditional decision rules are applicable. Stinnett and Paltiel (1996) show that a mixed integer programming framework provides a bridge between cost-eectiveness analysis and the assumption of non-constant returns to scale. However, their suggestion requires information on costs and eectiveness for all patient groups and all treatments in the whole health care system before the allocation problem can be solved. This approach then is extremely data hungry. Therefore traditional cost-eectiveness analysis of health care programs where the evaluation measure is presented as an incremental costeectiveness ratio has no role. In this paper we analyze whether cost-eectiveness analysis can also be used in the case of increasing returns to scale. We interpret cost-eectiveness analysis as a tool for estimating a cost function. The cost function does not give enough information to decide how resources should be allocated, but by applying a budget approach or a stipulated constant price approach as a decision rule it is possible to decide which treatment alternatives should be used in treating patient populations. With the constant price approach, traditional cost-eectiveness analysis also has a role and incremental cost-eectiveness ratios can be given a meaningful economic interpretation and can guide decision making as regards resource allocation. However, it is important to take into account the special nature of the capital cost in the estimations of incremental costeectiveness ratios. We start with a case with two independent patient groups which are homogeneous with respect
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G. Karlsson and M. Johannesson Table 1. Cost and eectiveness data for two independent patient groups Patient group 1 (P1) Number of patients: 1000
Treatment A B
c 100 200
e 10 11
Dc/De 10 100
to cost per patient treated and eectiveness per patient treated, with several treatment options. We then proceed by taking into account an investment program, which implies increasing returns to scale, that can be used for both of these patient groups. THE MODEL
Constant returns to scale In many cases an investment can be used for more than one patient group. In Table 1 variable cost per treatment (c), eectiveness per patient (e) (e.g. in terms of life-years or QALYs) and incremental cost-eectiveness ratios are shown for two independent patient groups. There are 1,000 patients in patient group 1 (P1) and 3,000 patients in patient group 2 (P2). In P1 there are two treatment options, A and B, and in P2 there is one treatment option, D. Within a patient group all treatment alternatives are mutually exclusive, which means that a patient can be treated with one and only one alternative. All patients within a patient group are homogeneous in that the cost and eectiveness per treatment are identical for all patients. The patient groups are independent in that the choice of treatment alternatives in one patient group does not aect the costs or eectiveness in the other patient group. All costs and eectiveness are compared to a baseline treatment, where cost and eectiveness are assumed to be zero. For all these treatments there exists no ®xed capital cost, so the assumption of constant returns to scale holds. The technology of treating these patients can be illustrated graphically. In Fig. 1 the aggregate costs (C) and aggregate eectiveness (E) for the two patient groups are shown separately*. If all patients in P1 are treated with A the aggregate costs amounts to 100 000 and the aggregate eectiveness to 10 000. By using combinations between the baseline treatment and treatment A it is possible to be on the solid curve between 0 and A (we disregard the fact that eectiveness has to be produced in steps of 10 in this interval as the eectiveness per treated patient is 10). If patients in patient group 1 *Aggregate cost (C) and aggregate eectiveness (E) denote the values for the whole population. {Alternative (X,Y) denotes that all patients in P1 receive treatment X and all patients in P2 receive treatment Y.
Patient group 2 (P2) Number of patients: 3000 Treatment D
c 100
e 5
Dc/De 20
(P1) are successively switched over from treatment A to treatment B, we move from point A towards point B along the solid curve, and when all patients are switched over to B we have reached point B. The same analysis applies also for patient group 2 (P2). Figure 1 can be interpreted as the aggregate cost curve of producing aggregate eectiveness for these patient groups with the available treatment options. What is the aggregate cost curve for treating P1 plus P2? If we start at zero cost and zero eectiveness the most ecient treatment is the one with the lowest incremental cost-eectiveness ratio, i.e. treatment A for P1. The next step is to add treatment D for P2 as this is the second lowest incremental costeectiveness ratio. When all patients in P1 receive treatment A and all patients in P2 are treated with treatment D, we are at point (A, D) in Fig. 1{. At point (B, D) all patients in P1 receive treatment B and all patients in P2 receive treatment D. Hence, the curve labelled P1 + P2 in Fig. 1 shows the aggregate cost curve for treating P1 plus P2. The analysis can also be performed in terms of the corresponding marginal cost and aggregate average cost functions. In Fig. 2 MC denotes the marginal cost and ATC the aggregate average cost of producing eectiveness. The marginal cost is the slope of the total cost curve, i.e. the curve (0)±(A, 0)±(A, D)±(B, D), in Fig. 1. The average cost is the slope of a curve from the origin to the total cost curve. As seen, the average total cost is equal to the marginal cost up to an aggregate eectiveness of 10 000 and thereafter the average total cost is lower than the marginal cost. The budget as the decision rule What are the most ecient treatment alternatives for these two patient groups? The decision rule can be based on two approaches (Karlsson and Johannesson, 1996). The ®rst is to determine the budget (or the aggregate cost) that we are prepared to spend, and the second is to determine the maximum price we are willing to pay to gain one unit of eectiveness. According to the budget approach the aggregate cost curve in Fig. 1 shows this relation. For example, if the budget is 100 000 all patients in P1 should receive treatment A and all patients in P2 should receive the baseline treatment, and with a budget of 400 000 all patients in P1 should receive treatment A and all patients in P2 should receive treatment D. If the budget amounts to 250 000 all
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Fig. 1. Cost function for patient group (P1) and patient group 2 (P2).
patients in P2 should receive treatment A, half of the patients in P2 should receive the baseline treatment and the rest of the patients in P2 should receive treatment D. The price as the decision rule Following the constant price approach, the alternative with the highest incremental cost-eectiveness ratio that is equal to or below the stipulated willingness to pay for one unit of eectiveness should be used (Karlsson and Johannesson, 1996). According to Fig. 2 the constant price approach means that the production of eectiveness should be increased as long as the stipulated constant price (p) exceeds MC, provided that p rATC. If, for example, p = 50 then all patients in P1 should be treated with alternative A and all patients in P2 with alternative D. Increasing returns to scale Now, let us assume that an investment in new equipment, I, can be used for the two patient
groups, P1 and P2. The ®xed capital cost, e.g. the user cost (UC) for one period, for the equipment is assumed to be 200 000. The variable cost per treatment is 200 for P1 and 90 for P2, respectively. The eectiveness per patient is 15 for P1 and 5 for P2. The implication of this new technology is that patient groups P1 and P2 are no longer independent. The choice of treatment in one patient group will aect the costs and eectiveness in the other patient group. The technologies with and without investment are illustrated graphically in Fig. 3. The solid curves show the cost function of producing aggregate eectiveness in P1 plus P2 with and without the investment. The curve (UC)±(I, 0)±(I, I) is the aggregate cost curve with the investment and (0, 0)±(A, 0)± (A, D)±(B, D) is the aggregate cost curve without the investment. It is possible to move successively along the solid cost curves, but it is not possible to move successively between the cost curves, e.g. between (A, D) and (I, I). The reason why it is impossible to move successively from, for example,
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Fig. 2. Marginal cost (MC) and average cost (ATC) of producing eectiveness.
(A, D) to (I, I) is that patients cannot be switched over from treatment A (patient group P1) and D (patient group P2) to treatment I without imposing the investment cost*. If we rule out all dominated treatments{ the aggregate cost function for P1 plus P2 is the cost curve (0, 0)±(A, 0)±(A, D)±(B, D)± (Z)±(I, I) in Fig. 3. Note that this aggregate cost curve is discontinuous between (B, D) and (Z) as well as non-concave in the section (A, D)±(B, D)± (Z)±(I, I). The technologies of producing eectiveness in patient groups P1 plus P2 are also illustrated in Table 2. The selection of alternatives is done in order of their incremental cost-eectiveness ratios for P1 plus P2, with and without investment. In the ``with investment'' technology the ®xed capital cost is included. Alternative Z is a linear combination between (I, 0) and (I, I), where all patients in P1 receive treatment I, 2200 patients in P2 receive treatment I and 800 patients in P2 receive the baseline treatment. *For example, a linear combination between (A,D) and (I,I) such that half of the patients receive A and D respectively and half of the patients treatment I will give an aggregate cost of 635 000 and an aggregate eectiveness of 27 500. This is actually above the curve between (A,D) and (I,I). {An alternative is dominated by another alternative if it is more costly without having more eectiveness.
The budget as a decision rule The following cost curve shows the maximum aggregate eectiveness that is obtainable with a speci®ed budget: curve (0, 0)±(A, 0)±(A, D)±(B, D)±(Z)±(I, I). For example, a budget of 625 000 for treating these two patient groups gives the maximum aggregate eectiveness of 27 500 when all patients in P1 are treated with treatment I, 2,500 patients in P2 are treated with I, and the remaining 500 patients in P2 are treated with the baseline treatment. This solution is a linear combination between (I, 0) and (I, I). Notice that this solution is located in the non-concave section of the aggregate cost function. This is the application of the budget approach, which shows the maximum aggregate eectiveness that is obtainable with a certain size of the budget. The price as a decision rule In the constant price approach a solution in the non-concave section of the aggregate cost function is not optimal. To see this we ®rst order all alternatives, both with the investment and without the investment, with respect to aggregate eectiveness. Then we calculate the incremental cost-eectiveness ratios, where the comparator is the alternative with the next best aggregate eectiveness. This is shown in Table 2, column 3. As can be seen from the table, the incremental cost-eectiveness ratio (DC/ DE) is decreasing when the alternative is changed
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Fig. 3. Cost function for P1 plus P2 with the investment and without the investment.
from (B, D) to (I, I). This means that alternative (B, D) is extended dominated by (A, D) and (I, I)*. Therefore alternative (B, D) should be excluded and the incremental cost-eectiveness ratios of the remaining alternatives should be recalculated. This is shown in Table 3. The decision rule is then that the alternative with the highest incremental costeectiveness ratio that is equal to or below the stipulated price should be chosen. For example, if the stipulated ®xed price is higher than 20 but below 54 alternative (A, D) should be chosen, and with a price equal to or higher than 54 alternative (I, I) is optimal. Notice that alternatives in the non-concave sector of the aggregate cost function are never optimal according to the price as a decision rule. This is contrary to the budget as a decision rule. *Extended dominance occurs when an alternative is dominated by a linear combination of two other alternatives. If mutually exclusive treatments are ranked according to their eectiveness extended dominance occurs if the incremental cost-eectiveness ratio decreases (Karlsson and Johannesson 1996). As linear combinations between (A,D) and (I,I) are not possible, this is not a case of extended dominance in the traditional sense.
The price as a decision rule can also be illustrated graphically. In Fig. 4 the marginal cost for the technology with the investment (MCw) and without (MCwo) is shown. The marginal cost that corresponds to Table 3 is labeled MCtot. MCtot coincides with MCwo up to 25 000 eectiveness units, where alternative (A, D) is used. Thereafter the marginal cost of producing eectiveness with the help of the technology without investment is 100. But it is possible to switch directly from (A, D) to (I, I); this produces another 5000 units of eectiveness at an average cost of 54 per unit. However, in this section the cost function exhibits indivisibilities; it is not possible to successively switch over from (A, D) to (I, I). If the price is equal to or exceeds 54 then alternative (I, I) is optimal; regardless of the price alternative (B, D) is always an inecient solution. In summary, an appropriate procedure for analyzing the cost-eectiveness in case of non-constant returns to scale is as follows: (1) de®ne all combinations that are mutually exclusive; (2) calculate aggregate costs and aggregate eectiveness for all these combinations; (3) rank the alternatives with respect to aggregate eectiveness (or aggregate costs); (4) calculate the appropriate incremental
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G. Karlsson and M. Johannesson Table 2. Aggregate cost (C) and aggregate eectiveness (E) for P1 + P2 1. Without investment
Alt 0, 0 A, 0 A, D B, D
C 0 100 000 400 000 500 000
E 0 10 000 25 000 26 000
3. With or without investment where dominated alternatives are excluded
2. With investment DC/DE Ð 10 20 100
Alt 0, 0 I, 0 I, I
C 200 000 400 000 670 000
E 0 15 000 30 000
DC/DE Ð 13.33 18
Alt 0, 0 A, 0 A, D B, D Z I, I
C 0 100 000 400 000 500 000 598 000 670 000
E 0 10 000 25 000 26 000 26 000 30 000
DC/DE Ð 10 20 100 Ð 18
Alt: Alternative; (X, Y) denotes treatment X for all patients in P1 and treatment Y for all patients in P2. Z is a linear combination between (I, 0) and (I, I), where all patients in P1 receive treatment I, 2200 patients in P2 receive treatment I and 800 patients in P2 receive baseline treatment.
cost-eectiveness ratios and exclude dominated and extended dominated alternatives. These incremental cost-eectiveness ratios guide the decision maker in a traditional way, provided that the constant price approach is acceptable. If there are many treatment options the number of mutually exclusive alternatives may be large. A method for keeping the number of combinations down is to ®rst calculate aggregate costs and aggregate eectiveness for each technology with a certain ®xed capital cost, such as the two technologies in Table 2 and Fig. 3. Thereafter all combinations can be ranked with respect to aggregate eectiveness. This method was used in the previous sections of this paper. Calculation of incremental cost-eectiveness ratio The evaluation measure between two relevant alternatives, such as (A, D) and (I, I) in Fig. 4, will be as follows. The incremental aggregate cost, formulated generally, for example when all patients are switched over from alternative (A, D) to (I, I), is DC DUC
m X Dcj *Nj
1
j1
where DC denotes the aggregate incremental cost of the program, DUC denotes the incremental ®xed capital cost, Dcj the incremental variable cost per patient for patient group j, Nj the number of patients in patient group j, and m the number of patient groups. The impact on aggregated eectiveness can be written: DE
m X Dej *Nj
2
j1
Table 3. Incremental cost-eectiveness ratios where dominated and extended dominated alternatives are excluded Alternative 0, 0 A, 0 A, D I, I a
C
E
DC/DEa
0 100 000 400 000 670 000
0 10 000 25 000 30 000
Ð 10 20 54
Compared to the next best alternative.
where DE denotes the increase in aggregate eectiveness, and Dej the incremental eectiveness per patient for patient group j. By dividing Equation (1) by Equation (2), dividing all terms by the total number of patients in all patient groups, and rearranging the terms we achieve: X X DUC= Nj kj Dcj DC j j X 3 DE kj Dej j
where kj=Nj/ajNj. Note that akj=1. Equation (3) shows the incremental cost-eectiveness ratio when the impacts of all costs and all health eects are taken into account for all patient groups. The ®rst term in the numerator shows the incremental capital cost per treatment, and the second term the sum of the variable incremental costs for all patient groups weighted by the share of patients in each patient group. The denominator shows the sum of incremental eectiveness for all patient groups where each patient group is weighted by its share of patients using the investment equipment. If the incremental cost-eectiveness ratio according to Equation (3) is equal to or below the stipulated constant price for one unit of eectiveness then the investment program should be implemented. It is seen from Equation (3) that the evaluation of the investment program has to take into account the impact on all patient groups (indications) simultaneously. For investment programs that are directed to one and only one patient group Equation (3) is simpli®ed to: DC IC=N Dc DE De
4
In this case the incremental cost (the numerator) is made up by the sum of the capital cost per treatment (IC/N) and the incremental variable cost per treatment, and the incremental eectiveness (the denominator) is the incremental eectiveness per treatment. In our numerical example the incremental costeectiveness ratio, when alternative (A, D) is com-
Cost-eectiveness analysis and capital costs
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Fig. 4. Marginal cost of producing eectiveness in P1 and P2 in the presence of increasing returns to scale.
pared to (I, I), is according to Equation (3): DC=DE f200000=4000 1=4* 200 ÿ 100 3=4* 90 ÿ 100g=f 1=4* 15 ÿ 10 3=4* 5 ÿ 5g 54 This is of course exactly the same result as in Table 3, where the two alternatives are compared. It may be tempting to simplify the evaluation of an investment program by calculating the cost-eectiveness ratio for each patient group (or indication) separately using Equation (4) instead of a simultaneous evaluation of the entire investment program for all patient groups according to Equation (3). It is easy to show that such a procedure will give incorrect solutions. For instance, let us use the numerical example and distribute the ®xed capital cost per treated patient ®rst assuming that the new equipment will be used for all the 4000 patients. The capital cost will then amount to 200 000/4000 = 50. For P2 the
cost of treatment I is then the capital cost plus the variable cost, i.e. 50 + 90 = 140, as compared to 100 for treatment D. With this method treatment D is less costly but gives the same eectiveness as treatment I. It seems to be the case that treatment I is dominated by treatment D and should therefore not be used. But this means in turn that only 1000 patients in P1 are potential users of treatment I and the capital costs increase to 200 000/1000 = 200 per patient. Applying this to P1, the incremental costeectiveness ratio amounts to {200 + (200 ÿ 100)}/ (15 ÿ 10) = 60. If the stipulated price is equal to 55 the conclusion is that treatment I should not be used for P2 either. With this method the investment should not be implemented at all if the ®xed price is equal to 55. But the previous analysis has shown that this is not correct. The correct conclusion is, as stated before, that if the stipulated price is equal to or above 54 the investment should be implemented and used for both P1 and P2. The reason why a
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separate evaluation of P1 and P2 leads to an incorrect conclusion is that the variable cost of using treatment I for P1 only is 90. If the new equipment is used for P1 then it is actually cost-saving to use this equipment for P2 as well. As seen, the cost for treating P2 is no longer independent of the choice of treatment alternatives for P1 and therefore the investment problem has to evaluate all potential patient groups simultaneously. DISCUSSION
Cost-eectiveness analysis can be interpreted in terms of calculating a cost function, where eectiveness is related to cost. Such a cost function is not enough to determine which alternatives should be used in treating patient populations. In order to do this, two types of decision rules can be applied to cost-eectiveness analysis: the budget approach and the constant price approach. So far, a key assumption in cost-eectiveness analysis has been the assumption of constant returns to scale. In the literature it has been claimed that this is a necessary assumption for cost-eectiveness analysis (Langley, 1996; Birch and Gafni, 1992, 1993), although Stinnett and Paltiel (1996) have pointed out that programming models based on the budget approach can also be used in cases of non-constant returns to scale. In this paper we have shown that a cost-eectiveness analysis framework can also be used in the presence of increasing returns to scale due to ®xed capital costs. Fixed capital costs create non-concavities in the cost function which may result in dierent solutions, depending on whether the budget approach or the constant-price approach is applied as the decision rule. Therefore the choice of decision rule may be important for the allocation of resources between competing health care programs. In cases of non-constant returns to scale the application of the budget as a decision rule for only parts of the health care sector, e.g. for speci®c patient groups, may result in suboptimization. If the budget approach is applied to the whole health care sector non-constant returns to scale may not be a major problem, if a speci®c patient group only makes up a small share of the whole patient population. However, there are risks of non-optimal solutions, but this is to a large degree an empirical issue. As Stinnett and Paltiel (1996) have shown, it is in principle possible to formulate an optimization problem based on the budget approach also for non-constant returns to scale. The major problem with the budget approach, however, is that it is extremely data hungry as it has to be applied to the whole health care sector simultaneously; information about costs and eectiveness of all potential treatments for all patient groups and indications is necessary. Therefore marginal conditions, such as incremental cost-eectiveness ratios, give no gui-
dance for decision making when the budget is used as a decision rule. This is probably the reason why Stinnett and Paltiel (1996) did not present any marginal conditions based on their formulations of the optimization problems Ð in our view it is hard to give marginal conditions any economically meaningful interpretation. Another problem with the budget approach is when a societal perspective is used. With a societal perspective a total societal cost has to be de®ned, but in such a case there are many budgets and for some resources, such as travel costs for the patient, there are no explicit budgets at all. Using only one speci®c budget, for example a hospital budget, means that all costs outside the hospital will be excluded. The inclusion of future costs will also create problems as a budget normally lasts for one year. Therefore for cost-eectiveness analysis to be of practical use at the moment in decisions regarding resource allocation within the health care sector, the decision rule has to be based on the constant price approach. However, there are weaknesses also with the constant price approach. Although there is a ``shadow budget'' to every stipulated constant price in theory, the complete aggregate cost function for the whole health care sector is unknown. In a world with uncertainty this means that a consistent application of the constant price approach as a decision rule will result in a total cost for health care that is actually unknown in advance. However, it is not only the valuation of health, measured as the price of eectiveness, that is of interest. The total cost for health care is also important. For example, the price as a decision rule assumes that eectiveness is valued at a constant price. Economic theory is traditionally based on the assumption of a decreasing marginal rate of substitution, which has the implication that a high production of eectiveness Ð and an associated high cost Ð will give a low social marginal valuation of eectiveness in terms of willingness to pay and vice versa. As the ``shadow budget'' is not known in advance the valuation of eectiveness may have to be adjusted, depending on the outcome of total costs. As shown before, the price approach together with increasing returns to scale implies indivisibilities. It is not optimal or maybe not even possible to implement a part of an investment program. If the cost for the investment program makes up only a minor share of the total health care costs this is not necessarily a problem; resources can be allocated from many other programs in the health care sector. This is probably the case in most real-life situations. The investment cost of a gamma knife amounts to about 30 million Swedish kronor. For a clinic or a hospital this is a signi®cant fraction of the total cost, but in relation to the total health care cost in Sweden Ð about 125 billion Swedish kronor a year Ð it is a small fraction. In our
Cost-eectiveness analysis and capital costs
opinion, investment programs that are regarded as costly account for only a small fraction of the cost of the health care system, and it is possible to allocate resources to such programs. The argument that programs with high investment costs make up only a minor fraction of the total cost of the health care sector may lead some people to the conclusion that investment costs and capital costs can also be regarded as variable costs, at least in the long run. We share this opinion, but this does not imply that capital costs can be treated as variable costs in cost-eectiveness analysis. As shown in the previous analysis, it is important to include all impacts on costs and eectiveness of the investment for all patient groups simultaneously. This holds true irrespective of the length of life of the investment. The fact that capital costs account for only a minor fraction of the health care cost does not contradict this conclusion. For some programs the capital cost is the major part of the treatment cost. For example in the treatment of patients with acoustic neurinoma with radiosurgery, the capital cost amounts to about 40% of the health care cost (van Roijen et al., 1997). Another example where the capital cost amounts to a signi®cant fraction of the total treatment cost is extracorporeal shockwave lithotripsy of uretarial stones and gallstones. The capital cost for the equipment accounts for about 45% of the total treatment costs in the treatment of gallstones and about 40% for uretarial stones (SBU, 1990). Likewise, for some diagnostic procedures, such as CT-scans and MRIs, the capital cost is high. In such cases it is very important to
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take into account the special nature of the capital costs when estimating the incremental cost-eectiveness ratios. In practice we think that high capital costs are most likely for programs in the hospital sector, and the ideas in this paper are thus especially relevant to that sector. REFERENCES
Birch, S. and Gafni, A. (1992) Cost eectiveness analysis: Do current decision rules lead us where we want to be? Journal of Health Economics 11, 279±296. Birch, S. and Gafni, A. (1993) Changing the problem to ®t the solution: Johannesson and Weinstein's (mis)application of economics to real world problems. Journal of Health Economics 12, 469±476. Johannesson, M. and Weinsten, M. C. (1993) On the decision rules of cost-eectiveness analysis. J. Health Econ. 12, 459±467. Karlsson, G. and Johannesson, M. (1996) The Decision Rules of Cost-eectiveness Analysis. PharmacoEconomics 9, 113±120. Langley, C. (1996) Cost Eectiveness and the Allocation of Therapies in a Treating Population. PharmacoEconomics 10, 93±98. van Roijen, L. et al. (1997) Costs and eects of microsurgery and radiosurgery in the treatment of acoustic neurinoma. Acta Neurochirugica, forthcoming. SBU (1990) Shock wave treatment of uretarial stones and gallstones. The Swedish Council on Technology Assessment in Health Care. (In Swedish). Stinnett, A. A. and Paltiel, A. D. (1996) Mathematical programming for the ecient allocation of health care resources. Journal of Health Economics 15, 641±653. Weinstein, M. C. and Zeckhauser, R. (1973) Critical ratios and ecient allocation. J. Public. Econ. 2, 147±157. Weinstein, M. C. (1990) Principles of cost-eective resource allocation in health care organizations. Int. J. Technol. Assess. Health Care 6, 93±105.
Soc. Sci. Med. Vol. 46, No. 9, pp. 1183±1191, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0277-9536(97)10046-6 0277-9536/98 $19.00 + 0.00
COST-EFFECTIVENESS ANALYSIS AND CAPITAL COSTS GOÈRAN KARLSSON* and MAGNUS JOHANNESSON Stockholm School of Economics, P.O. Box 6501, S-113 83 Stockholm, Sweden AbstractÐTraditionally, economic evaluations in terms of cost-eectiveness analysis are based, explicitly or implicitly, on the assumption of constant returns to scale. This assumption has been criticized in the literature and the role of cost-eectiveness as a tool for decision making has been questioned. In this paper we analyze the case of increasing returns to scale due to ®xed capital costs. Cost-eectiveness analysis is regarded as a tool for estimating a cost function. To this cost function two types of decision rules can be applied, the budget approach and the constant price approach. It is shown that in the presence of ®xed capital costs the application of these two decision rules to a speci®c patient group will give dierent results. The budget approach may lead to suboptimizations, while using the price as a decision rule will give optimal solutions. With ®xed capital costs and when an investment can be used for treating several patient groups, these groups are no longer independent. Therefore the cost-eectiveness analysis has to be performed simultaneously for all patient groups that are potential users of the investment. # 1998 Elsevier Science Ltd. All rights reserved Key wordsÐcost-eectiveness analysis, decision rules, increasing returns to scale, capital cost, indivisibilities, economic evaluation
INTRODUCTION
Traditionally, economic evaluations in terms of cost-eectiveness analysis are based, explicitly or implicitly, on the assumption of constant returns to scale. This assumption means that the cost per treated patient and the eectiveness per treated patient are independent of the size of the program. The decision rules of cost-eectiveness analysis have previously been stated in the literature (Weinstein and Zeckhauser, 1973; Weinstein, 1990; Johannesson and Weinsten, 1993; Karlsson and Johannesson, 1996). For many treatment programs the assumption of constant returns to scale can be justi®ed. However, there are programs that exhibit high investment costs, for example treatment with the gamma knife and radiation therapy. In these cases the capital cost for the equipment makes up a signi®cant share of the total cost, and at least up to a certain level the average cost per treated patient is decreasing with the size of the program. Therefore the assumption of constant returns to scale does not hold. There is no consensus in the literature as to whether or not the assumption of constant returns to scale is an acceptable approximation to the real world situation. The existence of non-constant returns to scale, among other concerns, has led Birch and Gafni to the conclusion that cost-eectiveness analysis is not a practical tool for decisions about resource allocation (Birch and Gafni, 1992; Birch and Gafni, 1993). Johannesson and Weinsten *Author for correspondence.
(1993), on the other hand, argue that the traditional decision rules are applicable. Stinnett and Paltiel (1996) show that a mixed integer programming framework provides a bridge between cost-eectiveness analysis and the assumption of non-constant returns to scale. However, their suggestion requires information on costs and eectiveness for all patient groups and all treatments in the whole health care system before the allocation problem can be solved. This approach then is extremely data hungry. Therefore traditional cost-eectiveness analysis of health care programs where the evaluation measure is presented as an incremental costeectiveness ratio has no role. In this paper we analyze whether cost-eectiveness analysis can also be used in the case of increasing returns to scale. We interpret cost-eectiveness analysis as a tool for estimating a cost function. The cost function does not give enough information to decide how resources should be allocated, but by applying a budget approach or a stipulated constant price approach as a decision rule it is possible to decide which treatment alternatives should be used in treating patient populations. With the constant price approach, traditional cost-eectiveness analysis also has a role and incremental cost-eectiveness ratios can be given a meaningful economic interpretation and can guide decision making as regards resource allocation. However, it is important to take into account the special nature of the capital cost in the estimations of incremental costeectiveness ratios. We start with a case with two independent patient groups which are homogeneous with respect
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G. Karlsson and M. Johannesson Table 1. Cost and eectiveness data for two independent patient groups Patient group 1 (P1) Number of patients: 1000
Treatment A B
c 100 200
e 10 11
Dc/De 10 100
to cost per patient treated and eectiveness per patient treated, with several treatment options. We then proceed by taking into account an investment program, which implies increasing returns to scale, that can be used for both of these patient groups. THE MODEL
Constant returns to scale In many cases an investment can be used for more than one patient group. In Table 1 variable cost per treatment (c), eectiveness per patient (e) (e.g. in terms of life-years or QALYs) and incremental cost-eectiveness ratios are shown for two independent patient groups. There are 1,000 patients in patient group 1 (P1) and 3,000 patients in patient group 2 (P2). In P1 there are two treatment options, A and B, and in P2 there is one treatment option, D. Within a patient group all treatment alternatives are mutually exclusive, which means that a patient can be treated with one and only one alternative. All patients within a patient group are homogeneous in that the cost and eectiveness per treatment are identical for all patients. The patient groups are independent in that the choice of treatment alternatives in one patient group does not aect the costs or eectiveness in the other patient group. All costs and eectiveness are compared to a baseline treatment, where cost and eectiveness are assumed to be zero. For all these treatments there exists no ®xed capital cost, so the assumption of constant returns to scale holds. The technology of treating these patients can be illustrated graphically. In Fig. 1 the aggregate costs (C) and aggregate eectiveness (E) for the two patient groups are shown separately*. If all patients in P1 are treated with A the aggregate costs amounts to 100 000 and the aggregate eectiveness to 10 000. By using combinations between the baseline treatment and treatment A it is possible to be on the solid curve between 0 and A (we disregard the fact that eectiveness has to be produced in steps of 10 in this interval as the eectiveness per treated patient is 10). If patients in patient group 1 *Aggregate cost (C) and aggregate eectiveness (E) denote the values for the whole population. {Alternative (X,Y) denotes that all patients in P1 receive treatment X and all patients in P2 receive treatment Y.
Patient group 2 (P2) Number of patients: 3000 Treatment D
c 100
e 5
Dc/De 20
(P1) are successively switched over from treatment A to treatment B, we move from point A towards point B along the solid curve, and when all patients are switched over to B we have reached point B. The same analysis applies also for patient group 2 (P2). Figure 1 can be interpreted as the aggregate cost curve of producing aggregate eectiveness for these patient groups with the available treatment options. What is the aggregate cost curve for treating P1 plus P2? If we start at zero cost and zero eectiveness the most ecient treatment is the one with the lowest incremental cost-eectiveness ratio, i.e. treatment A for P1. The next step is to add treatment D for P2 as this is the second lowest incremental costeectiveness ratio. When all patients in P1 receive treatment A and all patients in P2 are treated with treatment D, we are at point (A, D) in Fig. 1{. At point (B, D) all patients in P1 receive treatment B and all patients in P2 receive treatment D. Hence, the curve labelled P1 + P2 in Fig. 1 shows the aggregate cost curve for treating P1 plus P2. The analysis can also be performed in terms of the corresponding marginal cost and aggregate average cost functions. In Fig. 2 MC denotes the marginal cost and ATC the aggregate average cost of producing eectiveness. The marginal cost is the slope of the total cost curve, i.e. the curve (0)±(A, 0)±(A, D)±(B, D), in Fig. 1. The average cost is the slope of a curve from the origin to the total cost curve. As seen, the average total cost is equal to the marginal cost up to an aggregate eectiveness of 10 000 and thereafter the average total cost is lower than the marginal cost. The budget as the decision rule What are the most ecient treatment alternatives for these two patient groups? The decision rule can be based on two approaches (Karlsson and Johannesson, 1996). The ®rst is to determine the budget (or the aggregate cost) that we are prepared to spend, and the second is to determine the maximum price we are willing to pay to gain one unit of eectiveness. According to the budget approach the aggregate cost curve in Fig. 1 shows this relation. For example, if the budget is 100 000 all patients in P1 should receive treatment A and all patients in P2 should receive the baseline treatment, and with a budget of 400 000 all patients in P1 should receive treatment A and all patients in P2 should receive treatment D. If the budget amounts to 250 000 all
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Fig. 1. Cost function for patient group (P1) and patient group 2 (P2).
patients in P2 should receive treatment A, half of the patients in P2 should receive the baseline treatment and the rest of the patients in P2 should receive treatment D. The price as the decision rule Following the constant price approach, the alternative with the highest incremental cost-eectiveness ratio that is equal to or below the stipulated willingness to pay for one unit of eectiveness should be used (Karlsson and Johannesson, 1996). According to Fig. 2 the constant price approach means that the production of eectiveness should be increased as long as the stipulated constant price (p) exceeds MC, provided that p rATC. If, for example, p = 50 then all patients in P1 should be treated with alternative A and all patients in P2 with alternative D. Increasing returns to scale Now, let us assume that an investment in new equipment, I, can be used for the two patient
groups, P1 and P2. The ®xed capital cost, e.g. the user cost (UC) for one period, for the equipment is assumed to be 200 000. The variable cost per treatment is 200 for P1 and 90 for P2, respectively. The eectiveness per patient is 15 for P1 and 5 for P2. The implication of this new technology is that patient groups P1 and P2 are no longer independent. The choice of treatment in one patient group will aect the costs and eectiveness in the other patient group. The technologies with and without investment are illustrated graphically in Fig. 3. The solid curves show the cost function of producing aggregate eectiveness in P1 plus P2 with and without the investment. The curve (UC)±(I, 0)±(I, I) is the aggregate cost curve with the investment and (0, 0)±(A, 0)± (A, D)±(B, D) is the aggregate cost curve without the investment. It is possible to move successively along the solid cost curves, but it is not possible to move successively between the cost curves, e.g. between (A, D) and (I, I). The reason why it is impossible to move successively from, for example,
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Fig. 2. Marginal cost (MC) and average cost (ATC) of producing eectiveness.
(A, D) to (I, I) is that patients cannot be switched over from treatment A (patient group P1) and D (patient group P2) to treatment I without imposing the investment cost*. If we rule out all dominated treatments{ the aggregate cost function for P1 plus P2 is the cost curve (0, 0)±(A, 0)±(A, D)±(B, D)± (Z)±(I, I) in Fig. 3. Note that this aggregate cost curve is discontinuous between (B, D) and (Z) as well as non-concave in the section (A, D)±(B, D)± (Z)±(I, I). The technologies of producing eectiveness in patient groups P1 plus P2 are also illustrated in Table 2. The selection of alternatives is done in order of their incremental cost-eectiveness ratios for P1 plus P2, with and without investment. In the ``with investment'' technology the ®xed capital cost is included. Alternative Z is a linear combination between (I, 0) and (I, I), where all patients in P1 receive treatment I, 2200 patients in P2 receive treatment I and 800 patients in P2 receive the baseline treatment. *For example, a linear combination between (A,D) and (I,I) such that half of the patients receive A and D respectively and half of the patients treatment I will give an aggregate cost of 635 000 and an aggregate eectiveness of 27 500. This is actually above the curve between (A,D) and (I,I). {An alternative is dominated by another alternative if it is more costly without having more eectiveness.
The budget as a decision rule The following cost curve shows the maximum aggregate eectiveness that is obtainable with a speci®ed budget: curve (0, 0)±(A, 0)±(A, D)±(B, D)±(Z)±(I, I). For example, a budget of 625 000 for treating these two patient groups gives the maximum aggregate eectiveness of 27 500 when all patients in P1 are treated with treatment I, 2,500 patients in P2 are treated with I, and the remaining 500 patients in P2 are treated with the baseline treatment. This solution is a linear combination between (I, 0) and (I, I). Notice that this solution is located in the non-concave section of the aggregate cost function. This is the application of the budget approach, which shows the maximum aggregate eectiveness that is obtainable with a certain size of the budget. The price as a decision rule In the constant price approach a solution in the non-concave section of the aggregate cost function is not optimal. To see this we ®rst order all alternatives, both with the investment and without the investment, with respect to aggregate eectiveness. Then we calculate the incremental cost-eectiveness ratios, where the comparator is the alternative with the next best aggregate eectiveness. This is shown in Table 2, column 3. As can be seen from the table, the incremental cost-eectiveness ratio (DC/ DE) is decreasing when the alternative is changed
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Fig. 3. Cost function for P1 plus P2 with the investment and without the investment.
from (B, D) to (I, I). This means that alternative (B, D) is extended dominated by (A, D) and (I, I)*. Therefore alternative (B, D) should be excluded and the incremental cost-eectiveness ratios of the remaining alternatives should be recalculated. This is shown in Table 3. The decision rule is then that the alternative with the highest incremental costeectiveness ratio that is equal to or below the stipulated price should be chosen. For example, if the stipulated ®xed price is higher than 20 but below 54 alternative (A, D) should be chosen, and with a price equal to or higher than 54 alternative (I, I) is optimal. Notice that alternatives in the non-concave sector of the aggregate cost function are never optimal according to the price as a decision rule. This is contrary to the budget as a decision rule. *Extended dominance occurs when an alternative is dominated by a linear combination of two other alternatives. If mutually exclusive treatments are ranked according to their eectiveness extended dominance occurs if the incremental cost-eectiveness ratio decreases (Karlsson and Johannesson 1996). As linear combinations between (A,D) and (I,I) are not possible, this is not a case of extended dominance in the traditional sense.
The price as a decision rule can also be illustrated graphically. In Fig. 4 the marginal cost for the technology with the investment (MCw) and without (MCwo) is shown. The marginal cost that corresponds to Table 3 is labeled MCtot. MCtot coincides with MCwo up to 25 000 eectiveness units, where alternative (A, D) is used. Thereafter the marginal cost of producing eectiveness with the help of the technology without investment is 100. But it is possible to switch directly from (A, D) to (I, I); this produces another 5000 units of eectiveness at an average cost of 54 per unit. However, in this section the cost function exhibits indivisibilities; it is not possible to successively switch over from (A, D) to (I, I). If the price is equal to or exceeds 54 then alternative (I, I) is optimal; regardless of the price alternative (B, D) is always an inecient solution. In summary, an appropriate procedure for analyzing the cost-eectiveness in case of non-constant returns to scale is as follows: (1) de®ne all combinations that are mutually exclusive; (2) calculate aggregate costs and aggregate eectiveness for all these combinations; (3) rank the alternatives with respect to aggregate eectiveness (or aggregate costs); (4) calculate the appropriate incremental
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G. Karlsson and M. Johannesson Table 2. Aggregate cost (C) and aggregate eectiveness (E) for P1 + P2 1. Without investment
Alt 0, 0 A, 0 A, D B, D
C 0 100 000 400 000 500 000
E 0 10 000 25 000 26 000
3. With or without investment where dominated alternatives are excluded
2. With investment DC/DE Ð 10 20 100
Alt 0, 0 I, 0 I, I
C 200 000 400 000 670 000
E 0 15 000 30 000
DC/DE Ð 13.33 18
Alt 0, 0 A, 0 A, D B, D Z I, I
C 0 100 000 400 000 500 000 598 000 670 000
E 0 10 000 25 000 26 000 26 000 30 000
DC/DE Ð 10 20 100 Ð 18
Alt: Alternative; (X, Y) denotes treatment X for all patients in P1 and treatment Y for all patients in P2. Z is a linear combination between (I, 0) and (I, I), where all patients in P1 receive treatment I, 2200 patients in P2 receive treatment I and 800 patients in P2 receive baseline treatment.
cost-eectiveness ratios and exclude dominated and extended dominated alternatives. These incremental cost-eectiveness ratios guide the decision maker in a traditional way, provided that the constant price approach is acceptable. If there are many treatment options the number of mutually exclusive alternatives may be large. A method for keeping the number of combinations down is to ®rst calculate aggregate costs and aggregate eectiveness for each technology with a certain ®xed capital cost, such as the two technologies in Table 2 and Fig. 3. Thereafter all combinations can be ranked with respect to aggregate eectiveness. This method was used in the previous sections of this paper. Calculation of incremental cost-eectiveness ratio The evaluation measure between two relevant alternatives, such as (A, D) and (I, I) in Fig. 4, will be as follows. The incremental aggregate cost, formulated generally, for example when all patients are switched over from alternative (A, D) to (I, I), is DC DUC
m X Dcj *Nj
1
j1
where DC denotes the aggregate incremental cost of the program, DUC denotes the incremental ®xed capital cost, Dcj the incremental variable cost per patient for patient group j, Nj the number of patients in patient group j, and m the number of patient groups. The impact on aggregated eectiveness can be written: DE
m X Dej *Nj
2
j1
Table 3. Incremental cost-eectiveness ratios where dominated and extended dominated alternatives are excluded Alternative 0, 0 A, 0 A, D I, I a
C
E
DC/DEa
0 100 000 400 000 670 000
0 10 000 25 000 30 000
Ð 10 20 54
Compared to the next best alternative.
where DE denotes the increase in aggregate eectiveness, and Dej the incremental eectiveness per patient for patient group j. By dividing Equation (1) by Equation (2), dividing all terms by the total number of patients in all patient groups, and rearranging the terms we achieve: X X DUC= Nj kj Dcj DC j j X 3 DE kj Dej j
where kj=Nj/ajNj. Note that akj=1. Equation (3) shows the incremental cost-eectiveness ratio when the impacts of all costs and all health eects are taken into account for all patient groups. The ®rst term in the numerator shows the incremental capital cost per treatment, and the second term the sum of the variable incremental costs for all patient groups weighted by the share of patients in each patient group. The denominator shows the sum of incremental eectiveness for all patient groups where each patient group is weighted by its share of patients using the investment equipment. If the incremental cost-eectiveness ratio according to Equation (3) is equal to or below the stipulated constant price for one unit of eectiveness then the investment program should be implemented. It is seen from Equation (3) that the evaluation of the investment program has to take into account the impact on all patient groups (indications) simultaneously. For investment programs that are directed to one and only one patient group Equation (3) is simpli®ed to: DC IC=N Dc DE De
4
In this case the incremental cost (the numerator) is made up by the sum of the capital cost per treatment (IC/N) and the incremental variable cost per treatment, and the incremental eectiveness (the denominator) is the incremental eectiveness per treatment. In our numerical example the incremental costeectiveness ratio, when alternative (A, D) is com-
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Fig. 4. Marginal cost of producing eectiveness in P1 and P2 in the presence of increasing returns to scale.
pared to (I, I), is according to Equation (3): DC=DE f200000=4000 1=4* 200 ÿ 100 3=4* 90 ÿ 100g=f 1=4* 15 ÿ 10 3=4* 5 ÿ 5g 54 This is of course exactly the same result as in Table 3, where the two alternatives are compared. It may be tempting to simplify the evaluation of an investment program by calculating the cost-eectiveness ratio for each patient group (or indication) separately using Equation (4) instead of a simultaneous evaluation of the entire investment program for all patient groups according to Equation (3). It is easy to show that such a procedure will give incorrect solutions. For instance, let us use the numerical example and distribute the ®xed capital cost per treated patient ®rst assuming that the new equipment will be used for all the 4000 patients. The capital cost will then amount to 200 000/4000 = 50. For P2 the
cost of treatment I is then the capital cost plus the variable cost, i.e. 50 + 90 = 140, as compared to 100 for treatment D. With this method treatment D is less costly but gives the same eectiveness as treatment I. It seems to be the case that treatment I is dominated by treatment D and should therefore not be used. But this means in turn that only 1000 patients in P1 are potential users of treatment I and the capital costs increase to 200 000/1000 = 200 per patient. Applying this to P1, the incremental costeectiveness ratio amounts to {200 + (200 ÿ 100)}/ (15 ÿ 10) = 60. If the stipulated price is equal to 55 the conclusion is that treatment I should not be used for P2 either. With this method the investment should not be implemented at all if the ®xed price is equal to 55. But the previous analysis has shown that this is not correct. The correct conclusion is, as stated before, that if the stipulated price is equal to or above 54 the investment should be implemented and used for both P1 and P2. The reason why a
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separate evaluation of P1 and P2 leads to an incorrect conclusion is that the variable cost of using treatment I for P1 only is 90. If the new equipment is used for P1 then it is actually cost-saving to use this equipment for P2 as well. As seen, the cost for treating P2 is no longer independent of the choice of treatment alternatives for P1 and therefore the investment problem has to evaluate all potential patient groups simultaneously. DISCUSSION
Cost-eectiveness analysis can be interpreted in terms of calculating a cost function, where eectiveness is related to cost. Such a cost function is not enough to determine which alternatives should be used in treating patient populations. In order to do this, two types of decision rules can be applied to cost-eectiveness analysis: the budget approach and the constant price approach. So far, a key assumption in cost-eectiveness analysis has been the assumption of constant returns to scale. In the literature it has been claimed that this is a necessary assumption for cost-eectiveness analysis (Langley, 1996; Birch and Gafni, 1992, 1993), although Stinnett and Paltiel (1996) have pointed out that programming models based on the budget approach can also be used in cases of non-constant returns to scale. In this paper we have shown that a cost-eectiveness analysis framework can also be used in the presence of increasing returns to scale due to ®xed capital costs. Fixed capital costs create non-concavities in the cost function which may result in dierent solutions, depending on whether the budget approach or the constant-price approach is applied as the decision rule. Therefore the choice of decision rule may be important for the allocation of resources between competing health care programs. In cases of non-constant returns to scale the application of the budget as a decision rule for only parts of the health care sector, e.g. for speci®c patient groups, may result in suboptimization. If the budget approach is applied to the whole health care sector non-constant returns to scale may not be a major problem, if a speci®c patient group only makes up a small share of the whole patient population. However, there are risks of non-optimal solutions, but this is to a large degree an empirical issue. As Stinnett and Paltiel (1996) have shown, it is in principle possible to formulate an optimization problem based on the budget approach also for non-constant returns to scale. The major problem with the budget approach, however, is that it is extremely data hungry as it has to be applied to the whole health care sector simultaneously; information about costs and eectiveness of all potential treatments for all patient groups and indications is necessary. Therefore marginal conditions, such as incremental cost-eectiveness ratios, give no gui-
dance for decision making when the budget is used as a decision rule. This is probably the reason why Stinnett and Paltiel (1996) did not present any marginal conditions based on their formulations of the optimization problems Ð in our view it is hard to give marginal conditions any economically meaningful interpretation. Another problem with the budget approach is when a societal perspective is used. With a societal perspective a total societal cost has to be de®ned, but in such a case there are many budgets and for some resources, such as travel costs for the patient, there are no explicit budgets at all. Using only one speci®c budget, for example a hospital budget, means that all costs outside the hospital will be excluded. The inclusion of future costs will also create problems as a budget normally lasts for one year. Therefore for cost-eectiveness analysis to be of practical use at the moment in decisions regarding resource allocation within the health care sector, the decision rule has to be based on the constant price approach. However, there are weaknesses also with the constant price approach. Although there is a ``shadow budget'' to every stipulated constant price in theory, the complete aggregate cost function for the whole health care sector is unknown. In a world with uncertainty this means that a consistent application of the constant price approach as a decision rule will result in a total cost for health care that is actually unknown in advance. However, it is not only the valuation of health, measured as the price of eectiveness, that is of interest. The total cost for health care is also important. For example, the price as a decision rule assumes that eectiveness is valued at a constant price. Economic theory is traditionally based on the assumption of a decreasing marginal rate of substitution, which has the implication that a high production of eectiveness Ð and an associated high cost Ð will give a low social marginal valuation of eectiveness in terms of willingness to pay and vice versa. As the ``shadow budget'' is not known in advance the valuation of eectiveness may have to be adjusted, depending on the outcome of total costs. As shown before, the price approach together with increasing returns to scale implies indivisibilities. It is not optimal or maybe not even possible to implement a part of an investment program. If the cost for the investment program makes up only a minor share of the total health care costs this is not necessarily a problem; resources can be allocated from many other programs in the health care sector. This is probably the case in most real-life situations. The investment cost of a gamma knife amounts to about 30 million Swedish kronor. For a clinic or a hospital this is a signi®cant fraction of the total cost, but in relation to the total health care cost in Sweden Ð about 125 billion Swedish kronor a year Ð it is a small fraction. In our
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opinion, investment programs that are regarded as costly account for only a small fraction of the cost of the health care system, and it is possible to allocate resources to such programs. The argument that programs with high investment costs make up only a minor fraction of the total cost of the health care sector may lead some people to the conclusion that investment costs and capital costs can also be regarded as variable costs, at least in the long run. We share this opinion, but this does not imply that capital costs can be treated as variable costs in cost-eectiveness analysis. As shown in the previous analysis, it is important to include all impacts on costs and eectiveness of the investment for all patient groups simultaneously. This holds true irrespective of the length of life of the investment. The fact that capital costs account for only a minor fraction of the health care cost does not contradict this conclusion. For some programs the capital cost is the major part of the treatment cost. For example in the treatment of patients with acoustic neurinoma with radiosurgery, the capital cost amounts to about 40% of the health care cost (van Roijen et al., 1997). Another example where the capital cost amounts to a signi®cant fraction of the total treatment cost is extracorporeal shockwave lithotripsy of uretarial stones and gallstones. The capital cost for the equipment accounts for about 45% of the total treatment costs in the treatment of gallstones and about 40% for uretarial stones (SBU, 1990). Likewise, for some diagnostic procedures, such as CT-scans and MRIs, the capital cost is high. In such cases it is very important to
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take into account the special nature of the capital costs when estimating the incremental cost-eectiveness ratios. In practice we think that high capital costs are most likely for programs in the hospital sector, and the ideas in this paper are thus especially relevant to that sector. REFERENCES
Birch, S. and Gafni, A. (1992) Cost eectiveness analysis: Do current decision rules lead us where we want to be? Journal of Health Economics 11, 279±296. Birch, S. and Gafni, A. (1993) Changing the problem to ®t the solution: Johannesson and Weinstein's (mis)application of economics to real world problems. Journal of Health Economics 12, 469±476. Johannesson, M. and Weinsten, M. C. (1993) On the decision rules of cost-eectiveness analysis. J. Health Econ. 12, 459±467. Karlsson, G. and Johannesson, M. (1996) The Decision Rules of Cost-eectiveness Analysis. PharmacoEconomics 9, 113±120. Langley, C. (1996) Cost Eectiveness and the Allocation of Therapies in a Treating Population. PharmacoEconomics 10, 93±98. van Roijen, L. et al. (1997) Costs and eects of microsurgery and radiosurgery in the treatment of acoustic neurinoma. Acta Neurochirugica, forthcoming. SBU (1990) Shock wave treatment of uretarial stones and gallstones. The Swedish Council on Technology Assessment in Health Care. (In Swedish). Stinnett, A. A. and Paltiel, A. D. (1996) Mathematical programming for the ecient allocation of health care resources. Journal of Health Economics 15, 641±653. Weinstein, M. C. and Zeckhauser, R. (1973) Critical ratios and ecient allocation. J. Public. Econ. 2, 147±157. Weinstein, M. C. (1990) Principles of cost-eective resource allocation in health care organizations. Int. J. Technol. Assess. Health Care 6, 93±105.