- Email: [email protected]
A model of capitation
Journal
of Health
Economics
9 (1990) 397-409.
A MODEL
North-Holland
OF CAPITATION*
Thomas M. SELDEN Spcuse
University, Spraotse, NY 13244-1090,
Received
May
1989, final version
USA
received July 1990
This paper presents a theoretical model of capitation contracts. The consumer’s ex ante choice of medical plan is derived under flexible assumptions about provider-patient decision-making. The optimal medical plan is shown to combine full insurance with a provider payment system that is a mixture of capitation and partial reimbursement of provider costs. This solution strongly parallels the ‘mixed payment’ system derived by Ellis and McGuire (1986, 1990) in the context of prospective payment, though the optimal medical plan derived below may in fact be preferred to that solution in a world with endogenous admissions.
1. Introduction
An increasing proportion of medical care is financed with capitation, whereby the provider receives a fixed payment in return for supplying care ‘as needed’. Examples include (i) private health maintenance organizations (HMOs), and (ii) the Medicaid and Medicare experiments with capitation. Yet, despite its importance, there are to date no theoretical models of capitation.’ This paper is an attempt to develop such a model. The paper follows Ellis and McGuire (1986, 1990) in examining the consumer’s ex ante optimal choice of provider reimbursement and consumer insurance. The resulting ‘optimal medical plan’ is derived explicitly under certain simplifying assumptions. In particular, &e- kults lend support to Newhouse’s (1986) conjecture that the logic of Ellis and McGuire ‘mixed payment’ can be fruitfully extended to capitation. Indeed, in a model with endogenous admissions, capitation may be preferred to prospective payment, because capitation does not distort admissions decisions the way prospective payment might. Section 2 presents the formal model. Section 3 derives the optimal medical *This research was funded in part by National Institute of Mental Health Research Training Program, Department of Economics, University of Wisconsin-Madison. I thank Thomas Holmes and Burton -Weisbrod for their many cont;ibutions to this research. I also thank James Andreoni, Pius Eze. Thomas McGuire. Charles Meyer, Donald Morgan, Joseph Newhouse. Karl Scholz, Nancy Wolff, three anonymous referees, and seminar participants at S’yracuse University, University of Houston and University of Wisconsin-Madison for their helpful comments. ‘There are, of course, many insightful discussions of capitation. Recent examples include (but are not limited to) Anderson et al. (1986). England (1986), Freund and Neuschler (1986). McCall et al. (1987), and Newhouse (1986). 0167-6296/91/SO3.50
a
1991-Elsevier
Science
Publishers
B.V. (North-Holland)
398
T.&f. Selden, A model of capitation
plan under simplifying assumptions. Section 4 compares capitation and prospective payment systems. Section 5 examines issues surrounding the government’s use of capitation. Section 6 examines private HMO contracts. Section 7 examines the imperfect observability of provider costs and the role of nonprofit provision. Section 8 concludes the paper. 2. The model This section presents a one period model of a consumer who chooses a medical plan before illness severity, 13,is known. By medical plan, I mean an ex ante contract written between a consumer, a provider, and a third party payer, specifying both consumer insurance and provider payment. In the case of capitation, the medical plan may also include a stipulation that the provider furnish care ‘as needed’ to the consumer. The first-best case of no contractual limitations is examined first, followed by an analysis of medical care contracting in the presence of informational limitations on contracts. The consumer is assumed to maximize expected utility, &[U], where tIE [O, 11, with density function 4(e). Utility is assumed to be increasing, continuously differentiable, and concave in post-treatment health, h, and nonmedical consumption, y (the numeraire good): U = U(h(x, 0), y)
with
y= I-b,
(1)
where ~20 is the K-dimension vector of medical care, I is (lump-sum) income, and b is the individual’s medical bill (inclusive of any insurance premium).2 Post-treatment health is assumed concave in medical care (first increasing and then perhaps decreasing), with ah/CM CO and ?‘h/dx 20 > 0. For future reference, the consumer’s ex ante problem in a first-best world is to choose X, y, and b in each state of nature to solve maximize j u[h(x(e), e), y(em(e) r(.).~.).M.) fl subject
to
I -b(0)
Im
-y(e)
20
- ww4e~
d0
for all 0,
@a)
and
(2b)
PC)
deo,
where w is a vector of prices associated with treatment X, and where (2~) presumes the availability of actuarially-fair insurance. Assuming the indivi‘1 adopt the convention reimburses the provider.
that the consumer
pays all bills to the third-party
payer,
which then
T..Lf. Se&v.
4 model of capiration
399
dual chooses to consume some of the nonmedical good in every state of nature, the first-order necessary and sufficient conditions for a maximum are (i?U/i?h)(2h/2.~,) 5
i.**rv,
ZUlZy = y**(0),
and
(3b)
for all 19,
(3c)
y**(@=i.**
for all k (with strict equality
if .u:*(G)>O),
(3a)
where ;I(@ is the multiplier on constraint (2b), where i. is the multiplier on constraint (2c), where all derivatives are evaluated at (x**(e), y**(e), e), and where double asterisks denote first-best values. From (3~) we see that i.**( >O) can be viewed as the marginal utility of lump-sum income in the first-best consumer solution3 This solution maximizes consumer ex ante utility given w. If w is determined in competitive markets, and if there are no distortions in the market for I; then this solution is also efficient. Thus, an important issue is whether the consumer can choose a medical plan to attain this first-best solution within the contracting framework specified below. In contrast to the strong assumptions of the first-best model, the analysis below assumes that third-party insurers are unable to observe 8 ex post, so that insurance is of the form b=b(x*(@). In addition, the analysis follows Arrow (1963) in arguing that consumers may be unable to specify care ex ante, and may be less than fully sovereign ex post (perhaps because of imperfect information). In this case, we must carefully specify a model of provider-patient decision-making. This paper adopts the ‘agnostic’ assumption that x*(e) = argmax f(~(x),
h(x, e), I-b(x),
81,
(4)
where f(.) is assumed to be-increasing, continuously differentiable, and concave in provider net revenue, TC,and the arguments of patient utility (ft.) may also depend directly on 0). For simplicity it is assumed that y* = I - b(f+ye)) > 0. It is argued that x*(0) in (4) can be viewed as either (i) the treatment dictated by the provider in the case of supplier-induced demand, or (ii) the treatment agreed upon by the provider and the patient in a fully-informed bargaining process. In the case of supplier-induced demand f(.) can be interpreted as the provider’s objective function, assumed increasing in provider net revenue and the arguments of patient utility (where provider ‘caring’ may reflect either altruistic concerns for patient welfare or more ‘It may be helpful in this regard to note that the first-best involve the perfect shifting of income across states of nature x** =O).
consumer (including
solution will in general those states for which
400
TX.
Selden, A model of capitarion
selfish concerns over market share or malpractice litigation). Importantly, models of supplier-induced demand incorporate the case in which consumers have partial information, observing provider behavior with some ‘noise’. [Farley (1986) surveys this literature. See also Arrow (1963), Pauly and Satterthwaite (1981) and Ellis and McGuire (1986).] Alternatively, f(,) in eq. (4) might be viewed as a bargaining process between the provider and the fully-informed patient [Ellis and McGuire (1990)]. In this case, the equilibrium level of care will likely reflect both provider and patient interests. An interesting possibility in this case is that patient bargaining power may be a function of severity of illness.4 Provider net revenue in eq. (4) is assumed to be of the following form: Tc(x)=C+r(x)-wwx
(r-(0)=0),
where C is the capitation payment and r(.) is the provider reimbursement function. For instance, C >O and r(x) =0 for all x 10 is pure capitation, whereby the provider receives a fixed payment C and incurs all the costs of care. Pure fee-for-service is C =0 and &/2x, > 0 (for some k). Of particular interest in this paper are systems with C>O and 2r/t?xk>0, termed ‘mixed capitation and cost-based payment’. Combining these contractual assumptions, the consumer’s ex ante problem is to choose a medical plan that specifies consumer insurance and provider remuneration so as to provide both risk-sharing and appropriate incentives for ex post provider-patient decision-making: maximize
J U[h(+*(@, e), y*(@]$(Q de
subject to
y*(8) = I- b(x*(@)
for all 8,
(6)
and
x*(e) = argmax {(n(x), h(x, e), I-b(x), where Q is the consumer’s
@,
medical plan choice set.5 In general, R will
“If X* is viewed as a bargaining outcome satisfying individual rationality, it must be that P*zd, and U*zd,, where P* and U* denote the provider and consumer objectives evaluated at I’, and where d, and d, are the provider and consumer disagreement points. These conditions are assumed to hold, for simplicity. In partial justification of P*zd,, note that in a capitation contract d, may reflect the provider’s contractual obligation to supply care deemed necessary. In partial justification of U*zd,, note that I* can be understood to include second opinions and referrals sanctioned by the provider. Thus, consumer disagreement only occurs when the consumer refuses care recommended by the provider or purchases additional care from another provider. If ‘outside’ care is priced at w (or above), the optimal medical plan in section 3 will indeed ensure that no such disagreement occurs. ‘That is, Rc ‘Y x Y, where Y is the space containing all functions mapping the non-negative orthant of RK into R.
T.M. Selden, A model of capitation
401
depend on both insurer and provider interests. For instance, providers may be unwilling to provide care at an expected loss, and private insurers may be unwilling to offer medical plans at an expected loss. The ex ante consumer problem in (6) will in general be a highly complex optimal control problem. However, we can gain considerable insight by solving (6) under the simplifying assumptions detailed in the next section. 3. A simple case In this section, I make the following assumptions: (Al)
U=B(h(x,e))+I/(I-b(x)),
(A2) f(x)=~~(x)+ctB(h(x,8))+b1/(1_b(x))
with a>0 and 820,
(A3) 52= QF= the set of actuarially fair medical plans. That is, .QF={(rc(*),b(~))~EO[rr(~*(0))]=O and Ee[C+r(x*(8))-b(x*(@)]=O).
(7)
For instance, if 6=r in (A2), then f(.) might be viewed as a model of supplier-induced demand in which the provider values patient welfare at the rate of LY‘utils’ per ‘dollar’.6 Alternatively, f(.) in (A2) might represent a simple bargaining process, in which 6 is a measure of the patient’s bargaining power relative to the provider, while !x--6 measures the provider’s valuation of patient health benefit.’ Given (Al)-(A3), the following is a solution to the ex ante consumer’s problem in (6): (Sl)
b**(x)=h**
w
r**(x)
(S3)
C** = E,[wx**(@--r**(x**(@)],
=p**wx,
(=&CC**
+r**(x**(@)],
with p** = 1 --ai.**,
where A** is the marginal utility of lump-sum income from the consumer’s first-best problem (see (3~) above). This can be arbitrarily normalized to 6Alternatively, the provider might care only about patient health, ignoring the patient’s out-ofpocket expenses. In this case, z > 0 and 6 =O. ‘That is, f= P+dU, where P=n+(z-b)B is the provider’s objective function. Assuming an agreement is reached (see footnote 4) x*(0) is a bargaining solution satisfying (i) individual rationality, and (ii) independence of irrelevant alternatives. The solution does not permit afline transformations of either provider or patient utility, because the provider is assumed to care about B( .), and because of the linearity off(.) in I’(.) and U(.).
402
T..cI. Srlden, A model of capitarion
unity. Condition (Sl) states that full insurance is optimal, with the consumer paying a premium of 6** (set so that expected insurer profit is zero).’ Condition (S2) states that a constant fraction p** of all provider costs should be reimbursed by the insurer.’ And condition (S3) states that capitation should be set so that the provider breaks even. Indeed, we have the following result. Result 1. The medical consumer optimum.
plan
in (Sl)-(S2)
attains
the first-best
ex
ante
Proof. First, note that 6** and C** are chosen to satisfy the restriction that (I$.), b(.))EQF, so that (Sl)-(S3) is feasible. By definition, x*(e) solves
d f ldx, = drcldx, + r d B,ldx, + 6 d V/dx, 6 0 (with strict equality for x:(Q>O). sc(dB/8h)(dh/Zx,)s(
for all k
(8)
Substituting from (Sl) and (SZ), we have
1 -p**)wk
for all k
(9)
(with strict equality for x:(6)>0), since V(.) does not depend on x under full insurance. Substituting for p** yields (dB/dh)(dh/8x,)
5 w,i.**
for all k
(10)
(with strict equality for .u:(e)>O). Thus, x*(0)=x**(0) for all 8. Since treatment and risk sharing are both first best, (Sl)-(S3) solve the consumer’s (simplified) medical plan choice problem, replicating the first-best consumer optimum. 0 Moreover, if 4(@>0 for all eE[O, 11, we have that (SI)-(S3) is the unique solution to the simplified consumer problem.” *The optimality of full insurance results from the assumption of additively separable utility, so that it is not optimal to shift income across states of nature. This assumption is examined more fully below. 9A potential problem arises if p** ~0. This occurs if the patient has a strong bargaining position, if the provider places a strong weight on patient benefit, or if the consumer has a high marginal utility of non-medical consumption. In this case, the provider should be penalized for supplying care. Ellis and McGuire (1990) examine the case of an institutional constraint that ~20. In this case, the second-best solution will be p ** =0 , combined with incomplete consumer insurance. In practice, however, this case may be of secondary importance given the public concern over quality of care under full provider cost-sharing. ‘OFrom the concavity of B(.) and the uniqueness of x**(O), r**(.) is the unique r(.). such that r(0) =O and dr/&,
- wk + z( f?B/t?h)(Sh,‘Ex,) s 0
(with strict equality
at r**(0)
for all k,
e
for x:*(e) >O). C** and b** are then determined
uniquely
by (Sl) and (S3).
ZM.
403
Selden. A model of capitarion
There are several interesting aspects to this solution. First, reimbursement should be non-distortionary (i.e., provider cost-sharing, 1 -p**, is the same for all dimensions of care). Second, p** is a scalar less than one (for r >O), so that cost-plus payment is not optimal and C** > 0.’ ’ Third, optimal provider ‘cost-sharing’ depends positively on z. Thus, the analysis bears out Newhouse’s (1986, p. 53) conjecture that the logic of Ellis and McGuire ‘mixed payment’ can be fruitfully extended to medical plans based on capitation. Finally, optimal provider cost sharing will depend negatively on the consumer’s income. Consider for simplicity the case of K = 1. In this case, ds**(0)/dl~O, for all 19, with strict inequality for ,u**(@ ~0 (i.e.. health is a normal good). To achieve this increase in medical care, provider incentives wealthier individuals must be increased, so that dp**/dl >O.l’ Intuitively, demand medical plans that offer providers more generous incentives for the provision of care. This may be a partial answer to the question posed by Arrow (1963, p. 962) with respect to the factors expiaining participation in prepaid medical plans. Unfortunately, this solution does not hold in general if we weaken assumptions (Al)-(A3). For instance, one might wish to relax either (i) the assumption of separable utility (so that h affects marginal benefit from nonmedical consumption, and vice versa), or (ii) the assumption of lump-sum income (so that income might be a function of h). In either case, medical plans restricted to linear I**(.) will not in general be first best.” Another strong assumption is that f(,) is linear in xc, B, and I’. Nonlinearity of f(.) implies nonlinear reimbursement. Perhaps more importantly, 0 may enter f(.) directly. In a bargaining process, less severely ill patients may exert more bargaining power than more severely ill patients. “Thus, optimal provider payment can be viewed as a two-part tariff. Unlike pricing in decreasing cost industries, this two-part tariff ensures that the provider breaks even despite the financial disincentives at the margin. ‘*For KZ2 for all but one element of .c. - 3.‘c~may be an inferior good at 0 (i.e., dx:*(@,‘dl0 for some k= l,..., K). One can easily show that p must be increased to obtain an increase in n*, so that dp**,‘dl>O. 13When I depends on h, the first-best consumer solution is (~Li/Sh)(~h:~l,)+i.*‘[(ZI/Sh)(i;hlSI,)-w,]$O (with strict equality
for x:*(8) > 0) and
sujL?y = i.**, where all derivatives 2UjSy = j.**, provider
are evaluated reimbursement
Zr,‘&, -wL +zj(ZLi/3)(&‘Sx,)
at (x**(tI),y**(0),0). Even must still satisfy r(0) =O, and +~**[(~!/~h)(~hl’~.~~)
if b(.) can
be set
so
that
-(tb**,'?.~,)]) 50
(with strict equality for s:*(e)>O), where all derivatives are evaluated at (s**(0),y*‘(0), 0). Thus, r**(.) may well be nonlinear. (Note that the left-hand side of this inequality must be decreasing for all k to satisfy the second-order conditions for a provider-patient solution at x**(0).)
404
T..V. Selden, .A model of capiration
Or, in a model of supplier-induced demand, patients may know relatively little about the sophisticated treatments available for treatment of severe illness. In either case, r**(.) must be nonlinear if incentives are to adjust to the different weights placed on the arguments off(.). Perhaps the strongest assumption in the simple case is that the consumer chooses from the set of actuarially fair medical plans. In particular, if 4(.) is the private information of the consumer, there may be problems of adverse selection, with high-risk consumers pretending to be low-risk consumers, signing up for medical plans with low premia, full insurance, and generous incentives for the provision of care. In this case, coverage limitations on consumer insurance may arise to discriminate at least partially among risk types, while provider cost sharing may be reduced to lessen provider exposure.
4. A comparison with prospective payment Given (Al)-(A3), the medical plan in (Sl)-(S3) will be the unique first-best medical plan. Thus, medical plans utilizing prospective payment must be suboptimal. To see how prospective payment works in this model, let p>O be a prospective payment made conditional on x*zxmi”. For instance, in the context of in-patient hospital care xmi” might represent the minimum level of care that qualities as an ‘admission’. The resulting expression for provider net revenue can be written as follows: n(X) = C + pa(x) + r(x) - wx,
(11)
where a(.) is the admissions indicator function equal to one if x* zxmin and zero otherwise, and where we allow C_20 for generality. Because of the discontinuity at xmi”, we have an ‘admissions condition’: maxf>maxf 0 X2X”‘” I
x*zxmin
as well as the first-order admissions decision:
(12)
conditions
for treatment
for a = 1: d f /dx, s 0
(df/dx,
=0
for a=O:
(df/dx,=O
conditional
on the
if x: > x?‘“), (13)
df/d_x,sO
if x:>O).
Given this framework, we have the following result. Result 2.
If d(0) >O for all 9, then p>O is suboptimal.
T..Ll. Selden, A model of capitntion
405
Assume not, so that p** >O in an optimal medical Proof (for K= 1)” plan. Two necessary conditions for (n( .), 6( .)) to be optimal are (i) full insurance, and (ii) r(x)=p**wx. Define 19”“~(0, 11, such that ,~**(d”~“) =.Y”“. And define $E[O, l] such that for an arbitrary p, P+(P**-
1)w.K”i”+~B(h(x”i”,8))=(p**- 1)H**(8) +rB(h(.u**(8),
8)), (14)
so that the provider-patient solution exhibits ‘indifference’ with respect to admission. At p=O, x”~“=.Y**(~) with d=B”“, from (14). Totally differentiating (14) with respect to p and 8, usin g the envelope theorem and the properties of h(.), we have d&‘dp= {r[(dB,‘d&.)
-(dB/d$“,,)]I
- ’
where dB/del, denotes (?B/?h)(Sh/2B) evaluated B(0) = 0”” and S$/isp ~0. Thus, for p** >O x**(&p**))s**(e), which is a contradiction. probability given that #(@>O for all 0. 0
(15)
at s. That is, 8=$(p), with we have &p**) > emin, and emi”) are admitted, implying This occurs with positive
Intuitively, prospective payment will distort admissions decisions by offering incentives for the over-admission of low severity patients. Another concern about prospective payment is that providers may avoid treating unprofitable (i.e., high 0) patients.15 With capitation (or mixed capitation and cost-based payment), provider responsibility is established prior to realization of 8 (at least in theory). In the absence of capitation, providers may have far less responsibility for individuals they choose not to treat. In terms of a bargaining process, provider disagreement points may be different under the two scenarios, with prospective payment offering less severe penalties for providers choosing not to treat high 0 patients (see footnote 4). Thus, under assmptions (Al)-(A3), issues surrounding the endogeneity of admissions imply that optimal mixed capitation and cost-based reimbursement dominates payment systems using prospective payment. Outside of that special case, however, it may be second-best optimal to use prospective payments (perhaps in conjunction with capitation and cost-based reimbursement). For instance, in the case where 4(.) is private information and varies across individuals, prospective payments may help compensate providers for high-risk patients, while preserving treatment incentives conditional on admission. “Treating insight. “Dranove
x as a vector
complicates
(1987) and Pope (1989) address
the
proof
this issue.
considerably
without
yielding
additional
406
5. Government
T..Lf. Selden, .A model of capitarion
medical plans
In many cases, the government acts as the third-party payer, offering subsidized insurance to individuals and paying providers. In the simple model above, if capitation can be adjusted to reflect the risk types in a provider’s treatment population, then the government should use a mixture of capitation and cost-based payment - where the mixture reflects governmental concerns over quality of care. Indeed, Medicare has experimented with mixed capitation and cost-based reimbursement, whereby providers are permitted to keep a fraction of the difference between actual costs and a pre-determined threshold. l6 In addition, the states of Arizona, Missouri, Minnesota, New York, Oregon, and Wisconsin all currently employ pure capitation for at least a portion of their Medicaid populations. The adjustment of capitation to different treatment populations has proved difficult. Several Medicaid programs currently rely on competitive bidding among providers. Ideally, profit would be bid away for a particular beneficiary population given the government’s choice of I(*) and b(.). In practice, competitive bidding has been hampered by small numbers of providers [Christianson and Hillman (1986), Christianson et al. (1984), England (1986), McCall et al. (1987)]. Another strategy is to base capitation on the average charges (or some fraction thereof) in some other, fee-forservice sector of the medical-care market (typically the non-experimental beneficiary population). However, it is not clear to what extent these other populations match the experimental populations - particularly when selfselection across plans is allowed (and exercised) [Anderson et al. (1986), Freund and Neuschler (1986), Newhouse (1986)]. Attempts have been made to identify predictors of illness severity (or its imperfect, endogenous proxy utilization), so that capitation could be adjusted accordingly. Unfortunately, prediction has also proved problematic. To the extent reduced provider cost sharing (and, to a lesser extent, prospective payment) reduces systematic risk, the second-best medical plan may rely less heavily on capitation.
6. Private health maintenance contracts The contracts discussed above involve three parties: consumers, providers, and third-party payers (either private insurers or the government). With minor alterations, the model also depicts the more familiar contracts between a consumer and a health maintenance organization (HMO), in which there is no third party. In this case, the HMO receives a fixed fee, C, in return for 161n essence, provider net revenue in this case is n=p(ACwx) where AC is the cost threshold and ~~(0.1) is a sharing parameter (equal to 0.5 in the Medicare experiment). In order to clarify the relationship with mixed capitation and cost-based payment, this can be rewritten as n=C+(p-l)~xfor C=+4Cand p=I-p(tO.5).
T.&t. Selden, A model o)‘capitarion
407
supplying care ‘as needed’ to its members. where C is simply paid by the HMO member as an enrollment premium (b(O)=C). However, note that in the absence of a third-party payer, any fractional reimbursement of provider costs must be paid by the member as coinsurance. Formally, there is an additional constraint on the consumer’s problem that b(x)=C+r(x). In the absence of a third-party payer, this additional constraint will limit the ability of private health maintenance contracts to attain the ex ante consumer optimum described above (except in the case where p** =O). 7. Cost observability and non-profit provision A strong assumption in the analysis above is that payment can be specified as a function of cost (i.e., that wx is observable by third-party payers). In contrast, third parties are more likely to observe only certain aspects of care, such as length of stay, diagnostic tests performed, or surgical procedures performed, so that only these observable components of care can serve as a basis for reimbursement. Let ?cl be observable, while .Q is not observable. Then in the simple model under full insurance, we obtain the following from the first-order conditions for x*:
(16) so that the provision of care reflects the distortions inherent in the reimbursement system. Imperfect information of this form may offer a rationale for non-profit or public provision of hospital care. In the simple model above, if non-profit providers place a greater weight on patient health benefit than do for-profit providers (i.e., if aNeP>uFep), then P**~-~
8. Conclusion
This paper presents a framework
for thinking about a number of issues
“Related issues surrounding asymmetric information and the role of nonprofit provision are adclressed in Easley and O’Hara (1983), Hansmann (1980). and Weisbrod and Schlesinger (1986).
408
ZM.
Selden,
A model
of capitarion
surrounding the use of capitation. The paper derives the optimal medical plan under flexible assumptions about provider-patient decision-making. Under simplifying assumptions, the optimal medical plan is shown to combine full insurance for consumers with a provider payment system that is a mixture of capitation and fractional reimbursement of costs. Indeed, this medical plan is shown to be first best, dominating medical plans based on prospective payments in a model with endogenous admissions. These conclusions, however, are based on a number of strong assumptions - particularly with regard to the ability of third-party payers to observe the consumer’s risk type. An interesting result of weakening the assumption that costs are observable is that the optimal medical plan may incorporate non-profit provision. References Anderson, Cl.. J. Cantor, E. Steinberg and J. Holloway, 1986, Capitation pricing: Adjusting for prior utilization and physician discretion, Health Care Financing Review 8, 27-34. Arrow, K.. 1963, Uncertainty and the welfare economics of medical care, American Economic Review 53, 941-973. . Arrow, K., 1976, Welfare analysis of changes in health coinsurance rates, in: R. Rosett. cd., The role of health insurance in the health services sector (National Bureau of Economic Research, New York). Christianson, J. and D. Hillman, 1986. Health care for the indigent and competitive contracts: The Arizona experience (Health Administration Press, Ann Arbor, MI). Christianson, J., K. Smith and D. Hillman, 1984, A comparison of existing and alternative competitive bidding systems for indigent medical care, Social Science and Medicine 18, 599-604. Dranove, D., 1987, Rate-setting by diagnosis related groups and hospital specialization, Rand Journal of Economics 18, -tl7-427. Easley, D. and M. O’Hara. 1983, The economic role of the non-profit lirm, Bell Journal of Economics 14, 53 l-538. Ellis, R. and T. MC&ire, 1986, Provider behavior under prospective reimbursement: Costsharing and supply, Journal of Health Economics 5, 129-151. Ellis, R. and T. McGuire, 1990, Optimal payment systems for health services, Journal of Health Economics 9, 375-396 (this issue). England, W., 1986, Setting health maintenance organization capitation rates for Medicaid in Wisconsin, Health Care Financing Review 7, 67-73. Enthoven, A., 1980, Health plan: The only practical solution to the soaring cost of medical care (Addison-Wesley, Reading, MA). Farley, P., 1986, Theories of the price and quantity of physician services: A synthesis and critique, Journal of Health Economics 5, 3 15-333. Freund, D. and E. Neuschler, 1986, Overview of Medicaid capitation and case-management initiatives, Health Care Financing Review (suppl.), pp. 21-30. Hansmann, H., 1980, The role of nonprofit enterprises, Yale Law Journal 89, 835-901. Harris, J., 1979, Pricing rules for hospitals, Bell Journal of Economics 10, 224-243. McCall, N., D. Henton, S. Haber, L. Paringer, M. Crane, W. Wrightson and D. Freund, 1987, Evaluation of Arizona health care cost containment system, 1984-85, Health Care Financing Review 9, 79-90. McCuire, T., 1983, Patients’ trust and the quality of physicians, Economic Inquiry 21, 203-222. Newhouse, J., 1986, Rate adjusters for iMedicare under capitation, Health Care Financing Review (suppl.), pp. 45-55. Pauly, M.V., 1968, The economics of moral hazard, American Economic Review 58, 531-537.
T..M. Seiden, A model of capitation
409
Pauly, M.V., 1971, Indemnity insurance for health care efficiency, Economic Business Bulletin 24, 53-59. Pauly. M.V., 1980, Doctors and their workshops: Economic models of physician behavior (University of Chicago Press, Chicago, IL). Pauly, M.V. and M. Satterthwaite, 1981, The pricing of primary care physician’s service, Bell Journal of Economics 12.488-506. Pope, G., 1989, Hospital nonprice competition and Medicare reimbursement policy, Journal of Health Economics 8, 147-172. Reinhart, U., 1985. The theory of physician-induced demand: Reflections after a decade, Journal of Health Economics 4, 157-193. Roth, A., 1977, Independence of irrelevant alternatives, and solutions to Nash’s bargaining problem, Journal of Economic Theory 16, 247-251. Satterthwaite, M., 1979, Consumer information, equilibrium industry price, and the number of sellers, Bell Journal of Economics 10, 483-502. _ Selden. T.. 1988. Three essavs in oublic economics: Health care and famine relief. Ph.D. dissertation (University of Wisonsm-Madison). Shavell, S., 1979, Risk-sharing and incentives in the principal and agent relationship, Bell Journal of Economics 79, 55-73. Spence, M. and R. Zeckhauser, 1971, Insurance, information, and individual action, American Economic Review Proceedings 61, 380-387. Svejnar, J., 1986, Bargaining power, fear of disagreement, and wage settlements: Theory and evidence from U.S. industry, Econometrica 54, 1055-1078. Weisbrod, B. and M. Schlesinger, 1986, Public, private. nonprolit ownership and the response to asymmetric information: The case of nursing homes, in: S. Rose-Ackerman, ed., The economics of nonprofit institutions (Oxford University Press, New York), 133-151. Zeckhauser, R., 1970, IMedical insurance: A case study of the trade-off between risk spreading and appropriate incentives, Journal of Economic Theory 2, 10-26.
of Health
Economics
9 (1990) 397-409.
A MODEL
North-Holland
OF CAPITATION*
Thomas M. SELDEN Spcuse
University, Spraotse, NY 13244-1090,
Received
May
1989, final version
USA
received July 1990
This paper presents a theoretical model of capitation contracts. The consumer’s ex ante choice of medical plan is derived under flexible assumptions about provider-patient decision-making. The optimal medical plan is shown to combine full insurance with a provider payment system that is a mixture of capitation and partial reimbursement of provider costs. This solution strongly parallels the ‘mixed payment’ system derived by Ellis and McGuire (1986, 1990) in the context of prospective payment, though the optimal medical plan derived below may in fact be preferred to that solution in a world with endogenous admissions.
1. Introduction
An increasing proportion of medical care is financed with capitation, whereby the provider receives a fixed payment in return for supplying care ‘as needed’. Examples include (i) private health maintenance organizations (HMOs), and (ii) the Medicaid and Medicare experiments with capitation. Yet, despite its importance, there are to date no theoretical models of capitation.’ This paper is an attempt to develop such a model. The paper follows Ellis and McGuire (1986, 1990) in examining the consumer’s ex ante optimal choice of provider reimbursement and consumer insurance. The resulting ‘optimal medical plan’ is derived explicitly under certain simplifying assumptions. In particular, &e- kults lend support to Newhouse’s (1986) conjecture that the logic of Ellis and McGuire ‘mixed payment’ can be fruitfully extended to capitation. Indeed, in a model with endogenous admissions, capitation may be preferred to prospective payment, because capitation does not distort admissions decisions the way prospective payment might. Section 2 presents the formal model. Section 3 derives the optimal medical *This research was funded in part by National Institute of Mental Health Research Training Program, Department of Economics, University of Wisconsin-Madison. I thank Thomas Holmes and Burton -Weisbrod for their many cont;ibutions to this research. I also thank James Andreoni, Pius Eze. Thomas McGuire. Charles Meyer, Donald Morgan, Joseph Newhouse. Karl Scholz, Nancy Wolff, three anonymous referees, and seminar participants at S’yracuse University, University of Houston and University of Wisconsin-Madison for their helpful comments. ‘There are, of course, many insightful discussions of capitation. Recent examples include (but are not limited to) Anderson et al. (1986). England (1986), Freund and Neuschler (1986). McCall et al. (1987), and Newhouse (1986). 0167-6296/91/SO3.50
a
1991-Elsevier
Science
Publishers
B.V. (North-Holland)
398
T.&f. Selden, A model of capitation
plan under simplifying assumptions. Section 4 compares capitation and prospective payment systems. Section 5 examines issues surrounding the government’s use of capitation. Section 6 examines private HMO contracts. Section 7 examines the imperfect observability of provider costs and the role of nonprofit provision. Section 8 concludes the paper. 2. The model This section presents a one period model of a consumer who chooses a medical plan before illness severity, 13,is known. By medical plan, I mean an ex ante contract written between a consumer, a provider, and a third party payer, specifying both consumer insurance and provider payment. In the case of capitation, the medical plan may also include a stipulation that the provider furnish care ‘as needed’ to the consumer. The first-best case of no contractual limitations is examined first, followed by an analysis of medical care contracting in the presence of informational limitations on contracts. The consumer is assumed to maximize expected utility, &[U], where tIE [O, 11, with density function 4(e). Utility is assumed to be increasing, continuously differentiable, and concave in post-treatment health, h, and nonmedical consumption, y (the numeraire good): U = U(h(x, 0), y)
with
y= I-b,
(1)
where ~20 is the K-dimension vector of medical care, I is (lump-sum) income, and b is the individual’s medical bill (inclusive of any insurance premium).2 Post-treatment health is assumed concave in medical care (first increasing and then perhaps decreasing), with ah/CM CO and ?‘h/dx 20 > 0. For future reference, the consumer’s ex ante problem in a first-best world is to choose X, y, and b in each state of nature to solve maximize j u[h(x(e), e), y(em(e) r(.).~.).M.) fl subject
to
I -b(0)
Im
-y(e)
20
- ww4e~
d0
for all 0,
@a)
and
(2b)
PC)
deo,
where w is a vector of prices associated with treatment X, and where (2~) presumes the availability of actuarially-fair insurance. Assuming the indivi‘1 adopt the convention reimburses the provider.
that the consumer
pays all bills to the third-party
payer,
which then
T..Lf. Se&v.
4 model of capiration
399
dual chooses to consume some of the nonmedical good in every state of nature, the first-order necessary and sufficient conditions for a maximum are (i?U/i?h)(2h/2.~,) 5
i.**rv,
ZUlZy = y**(0),
and
(3b)
for all 19,
(3c)
y**(@=i.**
for all k (with strict equality
if .u:*(G)>O),
(3a)
where ;I(@ is the multiplier on constraint (2b), where i. is the multiplier on constraint (2c), where all derivatives are evaluated at (x**(e), y**(e), e), and where double asterisks denote first-best values. From (3~) we see that i.**( >O) can be viewed as the marginal utility of lump-sum income in the first-best consumer solution3 This solution maximizes consumer ex ante utility given w. If w is determined in competitive markets, and if there are no distortions in the market for I; then this solution is also efficient. Thus, an important issue is whether the consumer can choose a medical plan to attain this first-best solution within the contracting framework specified below. In contrast to the strong assumptions of the first-best model, the analysis below assumes that third-party insurers are unable to observe 8 ex post, so that insurance is of the form b=b(x*(@). In addition, the analysis follows Arrow (1963) in arguing that consumers may be unable to specify care ex ante, and may be less than fully sovereign ex post (perhaps because of imperfect information). In this case, we must carefully specify a model of provider-patient decision-making. This paper adopts the ‘agnostic’ assumption that x*(e) = argmax f(~(x),
h(x, e), I-b(x),
81,
(4)
where f(.) is assumed to be-increasing, continuously differentiable, and concave in provider net revenue, TC,and the arguments of patient utility (ft.) may also depend directly on 0). For simplicity it is assumed that y* = I - b(f+ye)) > 0. It is argued that x*(0) in (4) can be viewed as either (i) the treatment dictated by the provider in the case of supplier-induced demand, or (ii) the treatment agreed upon by the provider and the patient in a fully-informed bargaining process. In the case of supplier-induced demand f(.) can be interpreted as the provider’s objective function, assumed increasing in provider net revenue and the arguments of patient utility (where provider ‘caring’ may reflect either altruistic concerns for patient welfare or more ‘It may be helpful in this regard to note that the first-best involve the perfect shifting of income across states of nature x** =O).
consumer (including
solution will in general those states for which
400
TX.
Selden, A model of capitarion
selfish concerns over market share or malpractice litigation). Importantly, models of supplier-induced demand incorporate the case in which consumers have partial information, observing provider behavior with some ‘noise’. [Farley (1986) surveys this literature. See also Arrow (1963), Pauly and Satterthwaite (1981) and Ellis and McGuire (1986).] Alternatively, f(,) in eq. (4) might be viewed as a bargaining process between the provider and the fully-informed patient [Ellis and McGuire (1990)]. In this case, the equilibrium level of care will likely reflect both provider and patient interests. An interesting possibility in this case is that patient bargaining power may be a function of severity of illness.4 Provider net revenue in eq. (4) is assumed to be of the following form: Tc(x)=C+r(x)-wwx
(r-(0)=0),
where C is the capitation payment and r(.) is the provider reimbursement function. For instance, C >O and r(x) =0 for all x 10 is pure capitation, whereby the provider receives a fixed payment C and incurs all the costs of care. Pure fee-for-service is C =0 and &/2x, > 0 (for some k). Of particular interest in this paper are systems with C>O and 2r/t?xk>0, termed ‘mixed capitation and cost-based payment’. Combining these contractual assumptions, the consumer’s ex ante problem is to choose a medical plan that specifies consumer insurance and provider remuneration so as to provide both risk-sharing and appropriate incentives for ex post provider-patient decision-making: maximize
J U[h(+*(@, e), y*(@]$(Q de
subject to
y*(8) = I- b(x*(@)
for all 8,
(6)
and
x*(e) = argmax {(n(x), h(x, e), I-b(x), where Q is the consumer’s
@,
medical plan choice set.5 In general, R will
“If X* is viewed as a bargaining outcome satisfying individual rationality, it must be that P*zd, and U*zd,, where P* and U* denote the provider and consumer objectives evaluated at I’, and where d, and d, are the provider and consumer disagreement points. These conditions are assumed to hold, for simplicity. In partial justification of P*zd,, note that in a capitation contract d, may reflect the provider’s contractual obligation to supply care deemed necessary. In partial justification of U*zd,, note that I* can be understood to include second opinions and referrals sanctioned by the provider. Thus, consumer disagreement only occurs when the consumer refuses care recommended by the provider or purchases additional care from another provider. If ‘outside’ care is priced at w (or above), the optimal medical plan in section 3 will indeed ensure that no such disagreement occurs. ‘That is, Rc ‘Y x Y, where Y is the space containing all functions mapping the non-negative orthant of RK into R.
T.M. Selden, A model of capitation
401
depend on both insurer and provider interests. For instance, providers may be unwilling to provide care at an expected loss, and private insurers may be unwilling to offer medical plans at an expected loss. The ex ante consumer problem in (6) will in general be a highly complex optimal control problem. However, we can gain considerable insight by solving (6) under the simplifying assumptions detailed in the next section. 3. A simple case In this section, I make the following assumptions: (Al)
U=B(h(x,e))+I/(I-b(x)),
(A2) f(x)=~~(x)+ctB(h(x,8))+b1/(1_b(x))
with a>0 and 820,
(A3) 52= QF= the set of actuarially fair medical plans. That is, .QF={(rc(*),b(~))~EO[rr(~*(0))]=O and Ee[C+r(x*(8))-b(x*(@)]=O).
(7)
For instance, if 6=r in (A2), then f(.) might be viewed as a model of supplier-induced demand in which the provider values patient welfare at the rate of LY‘utils’ per ‘dollar’.6 Alternatively, f(.) in (A2) might represent a simple bargaining process, in which 6 is a measure of the patient’s bargaining power relative to the provider, while !x--6 measures the provider’s valuation of patient health benefit.’ Given (Al)-(A3), the following is a solution to the ex ante consumer’s problem in (6): (Sl)
b**(x)=h**
w
r**(x)
(S3)
C** = E,[wx**(@--r**(x**(@)],
=p**wx,
(=&CC**
+r**(x**(@)],
with p** = 1 --ai.**,
where A** is the marginal utility of lump-sum income from the consumer’s first-best problem (see (3~) above). This can be arbitrarily normalized to 6Alternatively, the provider might care only about patient health, ignoring the patient’s out-ofpocket expenses. In this case, z > 0 and 6 =O. ‘That is, f= P+dU, where P=n+(z-b)B is the provider’s objective function. Assuming an agreement is reached (see footnote 4) x*(0) is a bargaining solution satisfying (i) individual rationality, and (ii) independence of irrelevant alternatives. The solution does not permit afline transformations of either provider or patient utility, because the provider is assumed to care about B( .), and because of the linearity off(.) in I’(.) and U(.).
402
T..cI. Srlden, A model of capitarion
unity. Condition (Sl) states that full insurance is optimal, with the consumer paying a premium of 6** (set so that expected insurer profit is zero).’ Condition (S2) states that a constant fraction p** of all provider costs should be reimbursed by the insurer.’ And condition (S3) states that capitation should be set so that the provider breaks even. Indeed, we have the following result. Result 1. The medical consumer optimum.
plan
in (Sl)-(S2)
attains
the first-best
ex
ante
Proof. First, note that 6** and C** are chosen to satisfy the restriction that (I$.), b(.))EQF, so that (Sl)-(S3) is feasible. By definition, x*(e) solves
d f ldx, = drcldx, + r d B,ldx, + 6 d V/dx, 6 0 (with strict equality for x:(Q>O). sc(dB/8h)(dh/Zx,)s(
for all k
(8)
Substituting from (Sl) and (SZ), we have
1 -p**)wk
for all k
(9)
(with strict equality for x:(6)>0), since V(.) does not depend on x under full insurance. Substituting for p** yields (dB/dh)(dh/8x,)
5 w,i.**
for all k
(10)
(with strict equality for .u:(e)>O). Thus, x*(0)=x**(0) for all 8. Since treatment and risk sharing are both first best, (Sl)-(S3) solve the consumer’s (simplified) medical plan choice problem, replicating the first-best consumer optimum. 0 Moreover, if 4(@>0 for all eE[O, 11, we have that (SI)-(S3) is the unique solution to the simplified consumer problem.” *The optimality of full insurance results from the assumption of additively separable utility, so that it is not optimal to shift income across states of nature. This assumption is examined more fully below. 9A potential problem arises if p** ~0. This occurs if the patient has a strong bargaining position, if the provider places a strong weight on patient benefit, or if the consumer has a high marginal utility of non-medical consumption. In this case, the provider should be penalized for supplying care. Ellis and McGuire (1990) examine the case of an institutional constraint that ~20. In this case, the second-best solution will be p ** =0 , combined with incomplete consumer insurance. In practice, however, this case may be of secondary importance given the public concern over quality of care under full provider cost-sharing. ‘OFrom the concavity of B(.) and the uniqueness of x**(O), r**(.) is the unique r(.). such that r(0) =O and dr/&,
- wk + z( f?B/t?h)(Sh,‘Ex,) s 0
(with strict equality
at r**(0)
for all k,
e
for x:*(e) >O). C** and b** are then determined
uniquely
by (Sl) and (S3).
ZM.
403
Selden. A model of capitarion
There are several interesting aspects to this solution. First, reimbursement should be non-distortionary (i.e., provider cost-sharing, 1 -p**, is the same for all dimensions of care). Second, p** is a scalar less than one (for r >O), so that cost-plus payment is not optimal and C** > 0.’ ’ Third, optimal provider ‘cost-sharing’ depends positively on z. Thus, the analysis bears out Newhouse’s (1986, p. 53) conjecture that the logic of Ellis and McGuire ‘mixed payment’ can be fruitfully extended to medical plans based on capitation. Finally, optimal provider cost sharing will depend negatively on the consumer’s income. Consider for simplicity the case of K = 1. In this case, ds**(0)/dl~O, for all 19, with strict inequality for ,u**(@ ~0 (i.e.. health is a normal good). To achieve this increase in medical care, provider incentives wealthier individuals must be increased, so that dp**/dl >O.l’ Intuitively, demand medical plans that offer providers more generous incentives for the provision of care. This may be a partial answer to the question posed by Arrow (1963, p. 962) with respect to the factors expiaining participation in prepaid medical plans. Unfortunately, this solution does not hold in general if we weaken assumptions (Al)-(A3). For instance, one might wish to relax either (i) the assumption of separable utility (so that h affects marginal benefit from nonmedical consumption, and vice versa), or (ii) the assumption of lump-sum income (so that income might be a function of h). In either case, medical plans restricted to linear I**(.) will not in general be first best.” Another strong assumption is that f(,) is linear in xc, B, and I’. Nonlinearity of f(.) implies nonlinear reimbursement. Perhaps more importantly, 0 may enter f(.) directly. In a bargaining process, less severely ill patients may exert more bargaining power than more severely ill patients. “Thus, optimal provider payment can be viewed as a two-part tariff. Unlike pricing in decreasing cost industries, this two-part tariff ensures that the provider breaks even despite the financial disincentives at the margin. ‘*For KZ2 for all but one element of .c. - 3.‘c~may be an inferior good at 0 (i.e., dx:*(@,‘dl
for x:*(8) > 0) and
sujL?y = i.**, where all derivatives 2UjSy = j.**, provider
are evaluated reimbursement
Zr,‘&, -wL +zj(ZLi/3)(&‘Sx,)
at (x**(tI),y**(0),0). Even must still satisfy r(0) =O, and +~**[(~!/~h)(~hl’~.~~)
if b(.) can
be set
so
that
-(tb**,'?.~,)]) 50
(with strict equality for s:*(e)>O), where all derivatives are evaluated at (s**(0),y*‘(0), 0). Thus, r**(.) may well be nonlinear. (Note that the left-hand side of this inequality must be decreasing for all k to satisfy the second-order conditions for a provider-patient solution at x**(0).)
404
T..V. Selden, .A model of capiration
Or, in a model of supplier-induced demand, patients may know relatively little about the sophisticated treatments available for treatment of severe illness. In either case, r**(.) must be nonlinear if incentives are to adjust to the different weights placed on the arguments off(.). Perhaps the strongest assumption in the simple case is that the consumer chooses from the set of actuarially fair medical plans. In particular, if 4(.) is the private information of the consumer, there may be problems of adverse selection, with high-risk consumers pretending to be low-risk consumers, signing up for medical plans with low premia, full insurance, and generous incentives for the provision of care. In this case, coverage limitations on consumer insurance may arise to discriminate at least partially among risk types, while provider cost sharing may be reduced to lessen provider exposure.
4. A comparison with prospective payment Given (Al)-(A3), the medical plan in (Sl)-(S3) will be the unique first-best medical plan. Thus, medical plans utilizing prospective payment must be suboptimal. To see how prospective payment works in this model, let p>O be a prospective payment made conditional on x*zxmi”. For instance, in the context of in-patient hospital care xmi” might represent the minimum level of care that qualities as an ‘admission’. The resulting expression for provider net revenue can be written as follows: n(X) = C + pa(x) + r(x) - wx,
(11)
where a(.) is the admissions indicator function equal to one if x* zxmin and zero otherwise, and where we allow C_20 for generality. Because of the discontinuity at xmi”, we have an ‘admissions condition’: maxf>maxf 0 X2X”‘” I
x*zxmin
as well as the first-order admissions decision:
(12)
conditions
for treatment
for a = 1: d f /dx, s 0
(df/dx,
=0
for a=O:
(df/dx,=O
conditional
on the
if x: > x?‘“), (13)
df/d_x,sO
if x:>O).
Given this framework, we have the following result. Result 2.
If d(0) >O for all 9, then p>O is suboptimal.
T..Ll. Selden, A model of capitntion
405
Assume not, so that p** >O in an optimal medical Proof (for K= 1)” plan. Two necessary conditions for (n( .), 6( .)) to be optimal are (i) full insurance, and (ii) r(x)=p**wx. Define 19”“~(0, 11, such that ,~**(d”~“) =.Y”“. And define $E[O, l] such that for an arbitrary p, P+(P**-
1)w.K”i”+~B(h(x”i”,8))=(p**- 1)H**(8) +rB(h(.u**(8),
8)), (14)
so that the provider-patient solution exhibits ‘indifference’ with respect to admission. At p=O, x”~“=.Y**(~) with d=B”“, from (14). Totally differentiating (14) with respect to p and 8, usin g the envelope theorem and the properties of h(.), we have d&‘dp= {r[(dB,‘d&.)
-(dB/d$“,,)]I
- ’
where dB/del, denotes (?B/?h)(Sh/2B) evaluated B(0) = 0”” and S$/isp ~0. Thus, for p** >O x**(&p**))
(15)
at s. That is, 8=$(p), with we have &p**) > emin, and emi”) are admitted, implying This occurs with positive
Intuitively, prospective payment will distort admissions decisions by offering incentives for the over-admission of low severity patients. Another concern about prospective payment is that providers may avoid treating unprofitable (i.e., high 0) patients.15 With capitation (or mixed capitation and cost-based payment), provider responsibility is established prior to realization of 8 (at least in theory). In the absence of capitation, providers may have far less responsibility for individuals they choose not to treat. In terms of a bargaining process, provider disagreement points may be different under the two scenarios, with prospective payment offering less severe penalties for providers choosing not to treat high 0 patients (see footnote 4). Thus, under assmptions (Al)-(A3), issues surrounding the endogeneity of admissions imply that optimal mixed capitation and cost-based reimbursement dominates payment systems using prospective payment. Outside of that special case, however, it may be second-best optimal to use prospective payments (perhaps in conjunction with capitation and cost-based reimbursement). For instance, in the case where 4(.) is private information and varies across individuals, prospective payments may help compensate providers for high-risk patients, while preserving treatment incentives conditional on admission. “Treating insight. “Dranove
x as a vector
complicates
(1987) and Pope (1989) address
the
proof
this issue.
considerably
without
yielding
additional
406
5. Government
T..Lf. Selden, .A model of capitarion
medical plans
In many cases, the government acts as the third-party payer, offering subsidized insurance to individuals and paying providers. In the simple model above, if capitation can be adjusted to reflect the risk types in a provider’s treatment population, then the government should use a mixture of capitation and cost-based payment - where the mixture reflects governmental concerns over quality of care. Indeed, Medicare has experimented with mixed capitation and cost-based reimbursement, whereby providers are permitted to keep a fraction of the difference between actual costs and a pre-determined threshold. l6 In addition, the states of Arizona, Missouri, Minnesota, New York, Oregon, and Wisconsin all currently employ pure capitation for at least a portion of their Medicaid populations. The adjustment of capitation to different treatment populations has proved difficult. Several Medicaid programs currently rely on competitive bidding among providers. Ideally, profit would be bid away for a particular beneficiary population given the government’s choice of I(*) and b(.). In practice, competitive bidding has been hampered by small numbers of providers [Christianson and Hillman (1986), Christianson et al. (1984), England (1986), McCall et al. (1987)]. Another strategy is to base capitation on the average charges (or some fraction thereof) in some other, fee-forservice sector of the medical-care market (typically the non-experimental beneficiary population). However, it is not clear to what extent these other populations match the experimental populations - particularly when selfselection across plans is allowed (and exercised) [Anderson et al. (1986), Freund and Neuschler (1986), Newhouse (1986)]. Attempts have been made to identify predictors of illness severity (or its imperfect, endogenous proxy utilization), so that capitation could be adjusted accordingly. Unfortunately, prediction has also proved problematic. To the extent reduced provider cost sharing (and, to a lesser extent, prospective payment) reduces systematic risk, the second-best medical plan may rely less heavily on capitation.
6. Private health maintenance contracts The contracts discussed above involve three parties: consumers, providers, and third-party payers (either private insurers or the government). With minor alterations, the model also depicts the more familiar contracts between a consumer and a health maintenance organization (HMO), in which there is no third party. In this case, the HMO receives a fixed fee, C, in return for 161n essence, provider net revenue in this case is n=p(ACwx) where AC is the cost threshold and ~~(0.1) is a sharing parameter (equal to 0.5 in the Medicare experiment). In order to clarify the relationship with mixed capitation and cost-based payment, this can be rewritten as n=C+(p-l)~xfor C=+4Cand p=I-p(tO.5).
T.&t. Selden, A model o)‘capitarion
407
supplying care ‘as needed’ to its members. where C is simply paid by the HMO member as an enrollment premium (b(O)=C). However, note that in the absence of a third-party payer, any fractional reimbursement of provider costs must be paid by the member as coinsurance. Formally, there is an additional constraint on the consumer’s problem that b(x)=C+r(x). In the absence of a third-party payer, this additional constraint will limit the ability of private health maintenance contracts to attain the ex ante consumer optimum described above (except in the case where p** =O). 7. Cost observability and non-profit provision A strong assumption in the analysis above is that payment can be specified as a function of cost (i.e., that wx is observable by third-party payers). In contrast, third parties are more likely to observe only certain aspects of care, such as length of stay, diagnostic tests performed, or surgical procedures performed, so that only these observable components of care can serve as a basis for reimbursement. Let ?cl be observable, while .Q is not observable. Then in the simple model under full insurance, we obtain the following from the first-order conditions for x*:
(16) so that the provision of care reflects the distortions inherent in the reimbursement system. Imperfect information of this form may offer a rationale for non-profit or public provision of hospital care. In the simple model above, if non-profit providers place a greater weight on patient health benefit than do for-profit providers (i.e., if aNeP>uFep), then P**~-~
8. Conclusion
This paper presents a framework
for thinking about a number of issues
“Related issues surrounding asymmetric information and the role of nonprofit provision are adclressed in Easley and O’Hara (1983), Hansmann (1980). and Weisbrod and Schlesinger (1986).
408
ZM.
Selden,
A model
of capitarion
surrounding the use of capitation. The paper derives the optimal medical plan under flexible assumptions about provider-patient decision-making. Under simplifying assumptions, the optimal medical plan is shown to combine full insurance for consumers with a provider payment system that is a mixture of capitation and fractional reimbursement of costs. Indeed, this medical plan is shown to be first best, dominating medical plans based on prospective payments in a model with endogenous admissions. These conclusions, however, are based on a number of strong assumptions - particularly with regard to the ability of third-party payers to observe the consumer’s risk type. An interesting result of weakening the assumption that costs are observable is that the optimal medical plan may incorporate non-profit provision. References Anderson, Cl.. J. Cantor, E. Steinberg and J. Holloway, 1986, Capitation pricing: Adjusting for prior utilization and physician discretion, Health Care Financing Review 8, 27-34. Arrow, K.. 1963, Uncertainty and the welfare economics of medical care, American Economic Review 53, 941-973. . Arrow, K., 1976, Welfare analysis of changes in health coinsurance rates, in: R. Rosett. cd., The role of health insurance in the health services sector (National Bureau of Economic Research, New York). Christianson, J. and D. Hillman, 1986. Health care for the indigent and competitive contracts: The Arizona experience (Health Administration Press, Ann Arbor, MI). Christianson, J., K. Smith and D. Hillman, 1984, A comparison of existing and alternative competitive bidding systems for indigent medical care, Social Science and Medicine 18, 599-604. Dranove, D., 1987, Rate-setting by diagnosis related groups and hospital specialization, Rand Journal of Economics 18, -tl7-427. Easley, D. and M. O’Hara. 1983, The economic role of the non-profit lirm, Bell Journal of Economics 14, 53 l-538. Ellis, R. and T. MC&ire, 1986, Provider behavior under prospective reimbursement: Costsharing and supply, Journal of Health Economics 5, 129-151. Ellis, R. and T. McGuire, 1990, Optimal payment systems for health services, Journal of Health Economics 9, 375-396 (this issue). England, W., 1986, Setting health maintenance organization capitation rates for Medicaid in Wisconsin, Health Care Financing Review 7, 67-73. Enthoven, A., 1980, Health plan: The only practical solution to the soaring cost of medical care (Addison-Wesley, Reading, MA). Farley, P., 1986, Theories of the price and quantity of physician services: A synthesis and critique, Journal of Health Economics 5, 3 15-333. Freund, D. and E. Neuschler, 1986, Overview of Medicaid capitation and case-management initiatives, Health Care Financing Review (suppl.), pp. 21-30. Hansmann, H., 1980, The role of nonprofit enterprises, Yale Law Journal 89, 835-901. Harris, J., 1979, Pricing rules for hospitals, Bell Journal of Economics 10, 224-243. McCall, N., D. Henton, S. Haber, L. Paringer, M. Crane, W. Wrightson and D. Freund, 1987, Evaluation of Arizona health care cost containment system, 1984-85, Health Care Financing Review 9, 79-90. McCuire, T., 1983, Patients’ trust and the quality of physicians, Economic Inquiry 21, 203-222. Newhouse, J., 1986, Rate adjusters for iMedicare under capitation, Health Care Financing Review (suppl.), pp. 45-55. Pauly, M.V., 1968, The economics of moral hazard, American Economic Review 58, 531-537.
T..M. Seiden, A model of capitation
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