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WELFARE PROPERTIES OF
RESTRICTIONS TO HEALTH CARE
SERVICES BASED ON COST
EFFECTIVENESS
By
Laura Levaggi
Rosella Levaggi
Discussion Paper n. 0902
These Discussion Papers often represent preliminary or incomplete work, circulated to encourage discussion and
comments. Citation and use of such a paper should take account of its provisional character. A revised version
may be available directly from the author(s).
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Economiche, Università degli Studi di Brescia. Research disseminated by the Department may include views on
policy, but the Department itself takes no institutional policy position.
Welfare properties of restrictions to
health care services based on cost
e¤ectiveness
Laura Levaggi and Rosella Levaggiy
March 11, 2009
Abstract
In this note we explore the welfare properties of the cost e¤ectiveness criterion as an instrument to improve the appropriateness of
health care. We show that such instrument is optimal only under speci…c assumptions relating to the shape of the welfare function and the
utility of health care.
JEL Classi…cation: I11,I18
Key Words: Cost e¤ectiveness, welfare analysis
Department of Mathematics, University of Genova, Via Dodecaneso 35 - 16146 Genova
(Italy). E-mail: [email protected].
y
Department of Economics, University of Brescia, Via S. Faustino, 74b, 25122 Brescia
(Italy). E-mail: [email protected].
1
1
Introduction
In most countries, national authorities implement controls (e.g. budget impact limitations) and incentives (e.g. prescription limitations to be followed
by physicians) to in‡uence the use of health care. These systems, widely
used for drugs, are becoming more common for other types of treatments
(Jacobzone, 2000).
Restrictions on health care access are powerful instruments to reduce
demand. In their direct form (restriction on access or reimbursement) the
number of patients receiving a speci…c treatment is restricted to those that
are expected to derive the highest bene…ts.
The aim of this note is to study the welfare properties of direct restrictions based on cost-e¤ectiveness measurement from a theoretical point of
view, as a policy instrument that can be used to improve equity in access
to and …nancing of public health care systems. Its use increases the average
e¤ectiveness of care in a pure public health care system; the comparision
with a pure private health care system depends on the shape of utility.
Such criterion is welfare maximising only if the function representing utility
and the distribution of weights have speci…c properties. In general we can
therefore say that such criterion may increase average e¤ectiveness, but not
necessarily welfare.
2
The model
The community consists of N individuals, normalised to one. Each individual has a …xed exogenous income Y in the support (0; Q) and an endowment
H of health which produces utility according to a strictly increasing function
(Y ) with 0 (Y ) > 0.
This parameter depends upon individual characteristics such as education, income, social status and lifestyle. The utility function is additive and
separable in income and health; there is a …xed probability of being ill, if
so the health stock reduces to H. The utility function for the representative
consumer is:
(
y + (1
) (Y )H when healthy,
U=
(1)
y + (Y ) H;
if ill
where y and Y are the gross and net income respectively. Here captures
the greater concern of people with higher incomes with their health, due to
several characteristics: a greater opportunity cost of leisure and work time
spent in bad health, a greater concern for their physcal strength.
2
If ill, the patient can partially restore his health using a speci…c treatment that allows H units to be gained and has a cost equal to p1 .
is assumed to follow a known distribution on the support [0; 1
] with
2
g( ) > 0 representing its density.
3
Private provision
In a private health care market with no insurance, the patient buys the
treatment if he can a¤ord it (Y
p) and if the utility received is greater
p
than the cost. From (1) we can write (Y ) H
p, i.e.
(Y )H . Since
p
2 [0; 1
] the condition (Y )
H(1 ) is also necessary in order to
have a non-void interval of su¢ ciently high -s for convenience. To simplify
matters we assume that
p
(p)
;
(2)
H(1
)
so that any Y
p will satisfy this requirement, since is increasing. Therefore possibility/convenience conditions are
p
Y
p;
:
(3)
(Y )H
RQ
Let’s denote by E(Y ) = 0 Y f (Y ) dY the average income and by E( ) =
RQ
(Y ) f (Y ) dY the average .
0
In the private market the average e¤ectiveness of the drug will be equal
to
RQR1
g( ) d f (Y ) dY
p
p
(Y )H
pr
=
:
R
R
m
Q 1
g(
)
d
f
(Y
)
dY
p
p
(Y )H
For uniformly distributed
pr
m
and Y we get
RQh
)2
1 p (1
h
=
2 R Q (1
)
p
p2
)H 2
2 (Y
p
(Y )H
i
i
dY
:
dY
Such criterion may not be optimal both for e¢ ciency and equity reasons.
Patients buy the treatment on the basis of their income and e¤ectiveness.
Since the latter is just one of the determinants, productivity in terms of
health will never be maximised under this system.
1
To simplify the model, we assume that one unit of treatment is su¢ cient to treat the
patient.
2
For example, an active principle that reduces blood pressure might be less e¤ective if
used with other drugs, i.e. for a class of patients with multiple diseases.
3
4
The cost e¤ectiveness criterion
In a pure public health care system the treatment should in principle be
o¤ered to all those needing it. In this case, even if we assume that only
appropriate treatments are Rsupplied, the average e¤ectiveness of the new
1
treatment will be equal to 0
g( ) d . In order to improve the average
e¤ectiveness, most regulators use a simple cost e¤ectiveness criterion for
reimbursement:
p
p
<1
(4)
> ;
H
H
which in turn implies an expected average e¤ectiveness equal to:
R1
g( ) d
p
E
= RH1
;
g( ) d
p
H
which is always greater than the total average e¤ectiveness. For example in
the uniformly distributed case we obtain
E
1
=
2
p2
H2
p
H
(1
(1
)2
1hp
=
+ (1
)
2 H
i
)
pr
and E > 1 2 . In contrast, the sign of E
m depends on the shape of
(Y ). From a policy point of view, what is even more important is that we
do not know the welfare properties of such a criterion.
5
Welfare properties
Let’s examine the behaviour of a benevolent regulator who in this environment wishes to maximise social welfare. The derivation of social welfare
from individual preferences depends on the objectives pursued by the regulator. In general utility is assumed to be the sum, possibly weighted, of
individual utility3 . We use this approach here and de…ne total welfare as
follows:
Z QZ 1
V =
[y + (Y )H(1
+
+
(Y; ))] (Y )g( )d f (Y )dY:
0
0
The function (Y; ) 2 f0; 1g depends on the possibility/bene…t of the individual in being cured and re‡ects the choices of the regulator’s admission
3
For a discussion of the use of weights with a linear utility function see Fossati and
Levaggi (2007)
4
to the public market. The term
0 represents the weight the regulator
attaches to the utility of each individual. Such weights are positive, but decreasing in income, i.e. 0 (Y ) 0. From the above equation it is clear that,
without loss of generality, we can assume
Z Q
(Y ) f (Y ) dY = 1;
(5)
0
that is the function (Y ) := (Y )f (Y ) is a distribution on [0; Q].
5.1
Public provision
Let’s assume that the good can be supplied only by the public provider and
that the cost is …nanced through a linear income tax. The regulator de…nes
the minimum e¢ ciency level c over which the treatment is supplied, free of
charge. The optimal level has to be found to maximise the following welfare
function:
Z Q
Z Q
V = (1 t)
Y (Y )f (Y )dY + H (1
+ )
(Y ) (Y )f (Y )dY
0
+ H
Z
0
Q
(Y ) (Y )f (Y )dY
Z
(6)
1
g( )d ;
c
0
The budget constraint is
Z Q
Z
t
Y f (Y )dY = p
1
g( )d :
(7)
c
0
We introduce the notation
Z Q
E (Y ) :=
Y f (Y ) (Y ) dY;
E ( ) :=
0
Z
Q
(Y )f (Y ) (Y ) dY;
0
i.e. E (Y ) and E ( ) are the averages of income and
w.r.t. the new
distribution . As shown in Appendix A, the optimal value is:
c =
For
pE (Y )
:
HE ( )E(Y )
(8)
p
:
HE( )
(9)
= 1 we get
c =
From the above analysis we can derive the following propositions:
5
Proposition 1 For an utilitarian welfare function ( = 1) the cost e¤ ectiveness criterion (4) is optimal only if E( ) = 1. If this is not the case, the
average e¤ ectiveness is higher than E if E( ) < 1, lower when E( ) > 1.
Proposition 2 In a pure public health care system with weights on utility,
the cost e¤ ectiveness criterion is optimal if and only if
Z
0
Q
Y
f (Y ) (Y ) dY =
E(Y )
Z
Q
(Y )f (Y ) (Y ) dY:
(10)
0
Even in a context where there is no private market and the utility function
is linear in income, the cost e¤ectiveness criterion is optimal only under
special circumstances. For a system without weights on utility, using the cost
e¤ectiveness rule is equivalent to assuming that the average utility derived
from health care is equal to one; in a system where utility is weighted, the
Y
latter must have a particular shape. For a linear tax system E(Y
) is the
tax incidence function (Gouveia, 1997), i.e. the relative burden of each
individual. This allows interpretation of the equation (10) as follows: for
the cost e¤ectiveness criterion to be e¢ cient, the function of the weights
must equalise the average weighted “willingness to pay” (E ( )) with the
average weighted incidence of the …scal burden. If this is not the case, the
cost e¤ectiveness criterion is not optimal from a welfare point of view.
5.2
A mixed market
In most public health care systems the approval to commercialise the drug
is a process independent of reimbursement. In the presence of a prescription
limitation, patients whose productivity level is below c can still buy it on the
private market, provided that Y
p and p
(Y )H and
c. Therefore
buyers on the private market are identi…ed by this combination of Y and
p
(Y )H
c and p
Y
Q:
The number of buyers on the private market depends on c and we can identify a minimum income Y (c) (see (18) in Appendix B) such that consumers
satisfying
p
Y (p) Y
Q;
c
(Y )H
receive a utility surplus when receiving treatment. Using the notation of
Section 5.1, for a mixed health care market, the welfare function should be
6
written as:
W = (1
t)E (Y )
p
Z
Q
f (Y ) (Y )
Y (c)
+(1
+ H
+
Z
Q
Y (c)
Z
c
p
(Y )H
Z
)HE ( ) + HE ( )
!
Z
c
(Y )
p
(Y )H
g( ) d
!
dY
1
(11)
g( )d
c
g( )d
(Y )f (Y )dY
The budget constraint does not change, thus (7) is still valid. In Appendix
B we show that a necessary and su¢ cient condition for the existence of a
maximum is
Z Q
Z p
E (Y )
p
(Y ) dY < (1
)H
(Y ) (Y ) dY:
(12)
E(Y )
p
0
If this condition is satis…ed, from the F.O.C.
!
Z Q
Z Y (c)
p E (Y )
(Y ) dY = c
(Y ) (Y ) dY;
H E(Y )
Y (c)
0
(13)
by (5) we can write the optimal marginal reimbursed e¤ectiveness as:
!
R Y (c)
(Y
)
dY
p
E(Y
)
E
(Y
)
0
c=
1
:
(14)
R
R Y (c)
H Y (c) (Y ) (Y ) dY
E(Y )
(Y ) dY
0
0
The term
E ( ;Y
Y (c)) =
R Y (c)
0
(Y ) (Y ) dY
R Y (c)
(Y ) dY
0
is the average of under the distribution for individuals having income
Y
Y (c).
The expression (14) does not have a clear interpretation in its general
formulation. To gain some insights, let’s examine the case of a weight , for
which the averages E(Y ) and E (Y ) are equal 4 . We show in Appendix B
that in this case the optimal value is given by
p
HE ( ; Y
c = max
4
This condition is satis…ed whenever
uniformly distributed case f (Y ) =
1
Q
RQ
0
Y (1
p)
;1
(Y ))f (Y ) dY = 0; for example in the
a feasible choice is (Y ) =
7
:
6
Q
Y
Y2
Q
.
The expression E ( ; Y
p) is the average weighted utility of consumers
having insu¢ cient income to buy on the private market. This result is in
fact comparable to (9). In a pure public health care system, the regulator
chooses the marginal e¢ ciency level using the average level of , while in this
case only the utility loss of non-buyers is taken into account. If the regulator
gives higher weight to low income levels Y , then obviously E (Y ) < E(Y )
and the optimal c is lower.
Let’s now compare the cost e¤ectiveness criterion (4) with welfare maximisation in a mixed market. From (13) a minimum level Hp would be
optimal if and only if
Z Q
Z Y( p )
H
E (Y )
(Y ) dY =
(Y ) (Y ) dY
p
E(Y )
Y( )
0
H
Rearranging the terms, since by (18) (Y ( Hp )) = 1 we can show the following:
Proposition 3 In a mixed market case with weights on utility the cost effectiveness criterion is optimal if and only if
Z Q
Z Q
Y
(Y ) f (Y ) dY =
minf (Y ); 1g (Y ) f (Y )dY
(15)
0 E(Y )
0
Note that if (Y ) = 0 for Y
Y ( Hp ) conditions (15) and (10) coincide.
Observe also that the right-hand side in (15) is lower than E ( ). Thus,
if we compare a weight function ful…lling (15) with one satisfying (10), the
…rst has to give higher weight to incomes Y
E(Y ) than the second.
6
Discussion and conclusions
Cost e¤ectiveness measures are becoming increasingly popular among regulators as a tool to improve the appropriateness of health care services. Prescription limitations are an instrument used at various degrees by national
regulators. They may be e¤ective in increasing the expected e¤ectiveness
of a drug, but their welfare properties have not been explored so far. In
this paper we use a linear function very similar to the one proposed by the
literature (Jelovac, 2002) to show to what extent the cost e¢ cacy criterion
is also welfare maximising.
We show that in a pure public health care system where the drug is
available only through the public provider, cost e¢ cacy is also welfare maximising either if the utility derived from health has special properties or if
8
the set of weights is chosen so as to equalise average utility with average tax
incidence.
In a mixed market for health care, a straightforward comparison is more
di¢ cult because the marginal e¤ectiveness level that maximises welfare depends on and . In this case we might expect the cost e¤ectiveness criterion to be welfare improving only if the distribution of weights is skewed
towards low income brackets.
In general, we can say that the cost e¤ectiveness criterion is welfare
maximising only under speci…c conditions relating to the shape of these
two functions. Cost e¤ectiveness may be an easy to understand - easy to
implement criterion, but it does not necessarily imply that it is also welfare
maximising.
References
[1] Breyer, F. and A Hau‡er (2000) Health Care Reform: Separating Insurance from Income Redistribution, International Tax and Public Finance,
7( 4-5), pp. 445-61
[2] Fossati, A. and Levaggi, R. (2008) ,Delay is Not the Answer: Waiting Time in Health Care & Income Redistribution,
http://ssrn.com/abstract=1081928
[3] Jacobzone S (2000), Pharmaceutical policies in OECD: reconciling social
and industrial goals, OECD Labour market and social policy, occasional
papers.
[4] Jelovac, I (2002) On the relationship between negotiated prices of pharmaceuticals and patients’copayment, paper presented at the VI EHEW
meeting, Marseille, 8-9 November.
[5] Levaggi, L. and R. Levaggi (2006) Optimal copayment strategies in a
public health care system, http://ssrn.com/abstract=734923
[6] Smith, P (2005) User charges and priority setting in health care: balancing equity and e¢ ciency, Journal of Health Economics, 2005;24:10181029.
9
A
No private market
By de…ning the average level of
from (7) we have
t = t(c) =
as E( ) and the average income as E(Y ),
p
E(Y )
Z
1
g( )d :
(16)
c
Using the notation E (Y ) and E ( ) for the average of income and w.r.t.
the weighted distribution = f , we can write the objective function in
equation (6) as:
Z 1
V (c) = (1 t(c))E (Y ) + H (1
+ ) E ( ) + HE ( )
g( )d :
c
(17)
The optimization problem is the maximization of V on [0; 1
]. Since
p
dt
dc =
E(Y ) g(c), the …rst order condition for the optimization problem is
dV
E (Y )
= p
g(c)
dc
E(Y )
HE ( )cg(c) = 0
and the optimum value is
c =
pE (Y )
;
HE ( )E(Y )
which is independent of g.
B
Mixed market
Buyers on the private market are identi…ed by the conditions
p
(Y )H
c and p
Y
Q:
p
1
Inequality (Yp)H c is satis…ed if Y
cH : this lower bound is greater
p
p
than p when c
(p)H and greater than Q when c
(Q)H . From (2) we
p
always have (p)H
1
and we can de…ne
Y (c) :=
8
>
>
<Q
>
>
:p
0
1
p
cH
p
(Q)H
p
(p)H
10
p
(Q)H
c
p
(p)H
c
c
1
:
(18)
Let’s write the …rst order optimality conditions for the problem of maximizing the welfare function W ; to simplify notations write (Y ) = (Y )f (Y );
we have
Z Q
dt
dW
=
E (Y )
p
g(c) (Y )dY
HE ( ) c g(c)
dc
dc
Y (c)
Z Q
+ H
cg(c) (Y ) (Y ) dY
Y (c)
pE (Y )
=
g(c)
HE ( ) c g(c) + H
E(Y )
Z Q
p
g(c) (Y )dY:
Z
Q
cg(c) (Y ) (Y ) dY
Y (c)
Y (c)
Thus, using (5), we obtain
1 dW
=p
g(c) dc
E (Y )
E(Y )
Z
=p
E (Y )
E(Y )
1 +
Q
(Y ) dY
Y (c)
Z
!
Z
Y (c)
cH (Y ) (Y ) dY
0
Y (c)
(p
cH (Y )) (Y ) dY
(19)
0
We are interested in maximizing W on the interval [0; 1
], therefore we
have to analyze the sign and zeros of the above expression. The derivative
w.r.t. c of the right-hand side is
H
Z
Y (c)
(Y ) (Y ) dY;
0
thus W is concave. Because the right-hand side of (19) is positive for c = 0,
an optimum exists if and only if
Z Q
Z p
E (Y )
p
(Y ) dY
(1
)H
(Y ) (Y ) dY < 0
(20)
E(Y )
p
0
and in this case it satis…es the F.O.C.
!
Z Q
Z Y (c)
p E (Y )
(Y ) (Y ) dY:
(Y ) dY = c
H E(Y )
Y (c)
0
(21)
If the introduction of the weight does not in‡uence
h the averagei income, the
p
p
sign of the …rst derivative of W on the interval (Q)H
; (p)H
is positive.
11
In fact
(19)
is increasing, thus if Y
Z
dW
=
dc
Y (c) then by (18)
(Y )
p
cH ,
thus by
Y (c)
(p
cH (Y )) (Y ) dY > 0
0
and the optimal value can be found only if
p
c =
HE ( ; Y
h
lies in the interval
under the distribution
is increasing we have
p
(p)H ; 1
of
Z
i
Rp
(Y ) dY
p
Rp 0
=
p)
H 0 (Y ) (Y ) dY
. The term E ( ; Y
p) is the average
for individuals having income Y
p. Since
p
[ (p)
(Y )] (Y ) dY
0;
0
thus (p) E ( ; Y
value is given by
p) and c
c = max
the maximum point being 1
p
(p)H .
p
HE ( ; Y
Therefore in this case the optimal
p)
;1
if (20) is not satis…ed.
12
;
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DER WIEL, Erik WENGSTROM “Persuasion in Experimental Ultimatum Games” (luglio)
0812 – Rosella LEVAGGI “Decentralisation vs fiscal federalism in the presence of impure public
goods” (agosto)
0813 – Federico BIAGI, Maria Laura PARISI, Lucia VERGANO “Organizational Innovations
and Labor Productivity in a Panel of Italian Manufacturing Firms” (agosto)
0814 – Gianni AMISANO, Roberto CASARIN “Particle Filters for Markov-Switching StochasticCorrelation Models” (agosto)
0815 – Monica BILLIO, Roberto CASARIN “Identifying Business Cycle Turning Points with
Sequential Monte Carlo Methods” (agosto)
0816 – Roberto CASARIN, Domenico SARTORE “Matrix-State Particle Filter for Wishart
Stochastic Volatility Processes” (agosto)
0817 – Roberto CASARIN, Loriana PELIZZON, Andrea PIVA “Italian Equity Funds: Efficiency
and Performance Persistence” (settembre)
0818 – Chiara DALLE NOGARE, Matteo GALIZZI “The political economy of cultural
spending: evidence from italian cities” (ottobre)
Anno 2009
0901 – Alessandra DEL BOCA, Michele FRATIANNI, Franco SPINELLI, Carmine
TRECROCI “Wage Bargaining Coordination and the Phillips curve in Italy” (gennaio)