Marginal Costs of Life in Health Care:
Age, Gender and Regional Differences in Switzerland
Stefan Felder*,♣
April 2006
Abstract: An optimal allocation of health care expenditure over the life-cycle at each age sets
the value of life equal to the marginal costs of life. This paper estimates the age-specific
marginal costs of life using age-specific health care expenditures and mortality rates of Swiss
cantons for the period 1997 to 2003. It finds substantially higher marginal costs of life for
women compared to men, reflecting both a lower marginal productivity of medical inputs and
a higher spending for women. Medical technology does not differ across regions while
marginal costs of life do.
JEL Classification: I12, I18, J16
Keywords: Marginal costs of life, value of a statistical life, gender, health production,
regional diversity
*
♣
Correspondence to: Otto-von-Guericke University, Institute of Social Medicine and Health Economics,
Leipziger Str. 44, 39120 Magdeburg, Germany, Tel. +49 (0) 391 5328050, E-mail [email protected]
Thanks to Andreas Werblow for his help with the econometrics and discussions.
2
1. Introduction
In Switzerland as in other OECD countries the longevity of the population continues to rise.
Since 1960 the Swiss’ life-expectancy has increased by 9 years, from 72 to 81. Parallel to this
longevity gain the income share of health care has increased: while the average annual growth
rate of real income has been at 2 percent, health care expenditure has grown by 4 percent per
year (OECD, 2005).
There is a growing body of literature claiming a causality running from income to health care
expenditure to longevity: a rising income increases the value of life, which in turn increases
the demand for health care, which ultimately decreases mortality and increases longevity.1
The trace from increased health care expenditure to a lower mortality can be exemplified by
the change in technology for treating cardiovascular diseases. The introduction of angioplasty
in the early 80’s led to a substantial reduction in old age mortality and contributed almost half
of the total gain in life-years between 1980 and 2000 (see Cutler and McClellan (2001) and
Murphy and Topel (2005) for the USA and Felder (2006) for Germany).
In this paper I investigate the health care expenditure longevity channel using regional Swiss
data on age- and gender-specific health care expenditures and mortality rates. Switzerland is
an ideal candidate for analyzing medical technology as health care expenditures and life
expectancy significantly differ among the 26 cantons as does the income. By contrast, the
medical know-how, the education of physicians and the training of health care personnel are
similar across the regions, which lends support to a view that possible differences in the
marginal cost of saving a life are reflected by regional differences in the willingness to pay for
saving a life.
The paper is structured as follows: Section 2 presents the optimal allocation program of health
care expenditure across life-time, given at each age a technology relating health care
expenditures to the death rate. Section 3 gives information on the data used in this study.
Section 4 presents the empirical model and the results in two parts. The first part studies the
Swiss population as a whole, concentrating on differences in the marginal costs of life with
respect to age and gender. The focus of the second part is on regional differences of the
marginal costs of life. Section 5 summarizes and concludes.
1
See, among others, Hall and Jones (2005) and Becker et al. (2005).
3
2. Medical technology and marginal costs of life
Let us suppose that at each age the death rate is a function of health spending, and let the
technology be age-specific, to inhibit decreasing marginal returns and a constant input
elasticity:
qa = f a ( ma ) , with f a ' ( ma ) < 0 , f a '' ( ma ) > 0 and θ a ≡
f a ' ( ma ) ⋅ ma
,
f a ( ma )
(1)
where qa is the death rate and ma are health care expenditures at age a. Health spending is
assumed to influence the current mortality only. Still, it has an effect on the probability of
surviving through subsequent years. The probability of surviving from age a through age τ is
τ
Sa (τ ) = e
∫
− ft ( mt ) dt
a
,
(2)
so that ∂Sa (τ ) ∂ma = − f a ' ( ma ) ⋅ Sa (τ ) .
Furthermore let us assume that the flow of utility at age a depends on consumption, u = u ( ca )
with u ' ( ca ) > 0 and u '' ( ca ) < 0 , and ya is the income earned. Combining these assumptions
with that of the existence of a perfect annuity market, the maximization problem can be
written as
∞
max EU = ∫ S0 ( a ) ⋅ u ( ca ) da
ca , ma
0
s.t.
(3)
∞
∫ S (a) ⋅(c
0
a
+ ma − ya ) da = 0 .
0
The optimal choices of ca and ma satisfy the first-order conditions:
u ' ( ca ) = λ ,
∞
∫
a
⎡ ∞
⎤
f a ' ( ma ) ⋅ Sa (τ ) ⋅ u ( ca ) dτ = λ ⋅ ⎢1 + ∫ f a ' ( ma ) ⋅ Sa (τ ) ⋅ ( cτ + mτ − yτ ) dτ ⎥ .
⎣ a
⎦
(4)
(5)
(4) ensures that at each age the marginal utility of consumption is equalized. (5) states that the
expected utility gained through medical care at age a equals the expected costs related to that
expenditure. The expected utility increases since a drop in the death rate at age a changes all
conditional future survival probabilities. Costs have two components. The first is the direct
4
cost of health care. The second is the effect on net expenditures generated by the change in
survival probabilities because the residual life expectancy has changed. This is the additional
expenditure on consumption and medical care net of any earnings, as represented by the
second term on the right hand side of (5).
Rearranging (5) and using (4), one finds
∞
⎡ u ( ca )
⎤
1
MCLa ≡
= ∫ S a (τ ) ⋅ ⎢
− cτ − mτ + ya ⎥ dτ .
f a ' ( ma ) a
⎣ u ' ( ca )
⎦
(6)
The expression on the right hand side corresponds to the value of life at age a (see Rosen,
1988). The left hand side gives the additional spending needed to marginally decrease the
death rate at age a, i.e. the marginal costs of life. The optimal allocation sets the health care
spending at each age to equate the value of life to its marginal costs.
Using the definition of the elasticity of the death rate with respect to medical inputs (see (1)),
one arrives at
MCLa =
ma
.
θ a ⋅ f a ( ma )
(7)
According to (7), at each age the marginal costs of life are proportional to the health care
expenditure, and inversely related to the elasticity of the production function and the death
rate. When at age 80, the health care expenditure is 12,000 SFr., the death rate is 10 percent
and θ80 = 0.6 , the marginal costs of life are 400.000 SFr. By comparison, at age 30 the
average health care expenditure is at 2,000 SFr., the death rate is 0.05 percent, θ 30 = 1 ,
resulting in marginal costs of life equal to 4 million SFr.
Life is a normal good, in that an increase in wealth increases the value of life. This result of
comparative statics in the present model follows from the quasi-concavity of the utility
function (see Jones-Lee, 1976). A wealth increase decreases the marginal utility of wealth,
which means that more wealth will be sacrificed for a given decrease in the death rate.
3. Data
This study is based on age-profiles of health care expenditure and death rates in the 26 Swiss
cantons for the years 1997 to 2003. Health care data come from the social health insurance
system financed through regionally differentiated premiums and from public spending
financed by tax revenues on the cantonal and communal level. While the range of services
5
covered by social health insurance is fixed at the federal level, regional premiums differ
according to the local demand for health care. Community rating applies, i.e. premiums must
not differ with respect to age, sex or any other risk factors. Public spending mainly for inpatient care also reflects the regional population’s willingness to pay for health care services.
The social health insurance expenditure data distinguishes 5-year age intervals of health
spending for men and women, covering a life span of 100 years. I allocated public health
spending, which is not age-specific, to men and women and across age-intervals in proportion
to the observed expenditure profiles in social health insurance. Death rates are also available
in 5-year age intervals. I have adjusted the rates for exogenous death causes.2
Switzerland’s regions are heterogeneous with respect to the population’s longevity, health
care expenditure and income as can be gathered from Table 1. The life expectancy is highest
in the French and Italian speaking cantons. The population in the West also incurs the highest
level of health care expenditure. In terms of per capita income, the West ranges at the lower
end of the distribution. Zurich has the highest per capita income but shows average figures in
terms of longevity and health care expenditures. The shortest life expectancy is observed in
the central and eastern parts of Switzerland, where the average the health care expenditure is
low. The Center has higher average income than the East. Both the Midlands and the NorthWest feature average figures in all respects.
Table1: Population, life expectancy, per capita health care expenditures (HCE) and
income in the six Swiss regions, 1997 and 2003.
Population
(1,000,000)
Life expectancy
(years)
Per capita HCE
(1,000 SFr.a)
Per capita income
(1,000 SFr.)
Region
2003
1997
2003
1997
2003
1997
2003
West
1.67
79.75
81.01
3.40
4.52
35.86
40.05
Midlands
1.68
79.04
80.41
2.51
3.66
35.15
40.08
North-West
1.01
79.47
80.70
2.66
3.41
40.51
46.95
Center
0.70
79.52
81.00
2.10
2.93
39.22
43.92
East
1.06
78.90
80.40
1.96
2.76
34.74
41.74
Zurich
1.25
79.11
80.72
2.48
3.55
46.21
52.32
Switzerland
7.36
79.31
80.71
2.61
3.61
38.22
43.70
a
2
A Swiss Frank buys for 0.67 €.
Following ICD10 standards, codes V01-999 cover exogenous death causes, including accidents, suicides,
homicide, complications with medical treatments or surgery.
6
4. Quantitative analysis
The empirical part closely follows the approach taken by Hall and Jones (2005). Let the ageand time-specific medical technology be represented by a Cobb-Douglas production function:
qa ,t = γ a ⋅ ( ma ,t )
−θ a
,
(8)
where γ a is an efficiency parameter and θ a is the elasticity of the death rate with respect to
medical inputs. Both parameters of the production function are allowed to depend on age.
Any age-specific mortality rate will not only depend on medical spending but also on
unobserved variables such as education and the quality of the environment (air, water
pollution etc.). Moreover, the technical progress in the health sector will affect the efficiency
of medical inputs. The unobserved variables have a time trend as does the observed variable
health care expenditure. Estimating the production function straight from (8) resulted in
biased coefficients, since ma ,t would be correlated with the error term. One way to remove
the omitted-variable bias is to use a linear trend as an instrument and to specify a two stage
approach. On the first stage I regressed log ma ,t on the exogenous variables of the system
(age-specific constants and time) and got a predictor of ma ,t . This predictor, log m a ,t , was then
used on the second stage:
log qa ,t = log γ a − θ a ⋅ log m a ,t + ε a ,t ,
(9)
where ε a ,t is assumed to be a mean-zero, trend-less zero-term. I employed GMM to estimate
the age-specific production efficiency and elasticity parameters, γ a and θ a , resulting in small
standard errors.
4.1 Age and gender differences in the marginal costs of life
In the first set of estimates the focus is on the national level, using 5-year age intervals
differentiated with respect to men and women. Equation (9) was estimated separately for men
and women. In 2003 the average health care expenditure of a woman was one third higher
than that of a man (3,054 SFr. against 4,164 SFr. when public health spending is included).
Table 2 presents the results for the elasticity of the death rate with respect to health spending
and the marginal costs of life, its value in 2003 and its annual growth rate between 1997 and
2003. For infants and young children up to 10 years, the elasticity is not significantly different
7
from zero. This may have to do with the fact that the death rate at very young ages is very
small, given that a large progress in reducing the infant mortality was realized in earlier
decades (see Murphy and Topel, 2005, and Felder, 2006).
Table 2:
The elasticity of the death rate with respect to health care expenditure (θ),
and the marginal costs of life (MCL) of men and women by 5-yeas intervals
Age
Men
θ
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95+
0-95+
0-95+a
0.49
0.30
1.08**
1.30*
2.87**
1.79**
1.32**
0.60*
0.98**
0.61**
0.35**
0.42**
0.76**
0.57**
0.68**
0.53**
0.67**
0.48*
0.93**
0.29
0.93**
0.80**
MCL 2003
Growth rate
(in 1,000 SFr.) 1997-2003
2,450
5.16%
44,106
3.11%
10,952
8.46%
6,259
9.40%
1,879
13.57%
2,176
8.71%
2,562
8.25%
4,479
6.48%
1,960
7.17%
2,152
7.84%
2,590
5.29%
1,677
6.78%
703
6.74%
730
7.68%
436
6.92%
395
7.02%
214
6.68%
217
5.30%
84
3.74%
179
2.65%
5,304
4.51%
4,277
5.78%
Women
θ
0.23
1.75
1.42**
0.89**
1.51**
0.85*
0.95**
1.02**
0.89**
0.48*
0.75**
0.19*
0.29**
0.40**
0.44**
0.47**
0.63**
0.27**
0.40**
0.47**
0,82**
0.64**
Growth rate
MCL 2003
(in 1,000 SFr.) 1997-2003
5,439
4.56%
8,742
7.14%
10,729
8.95%
15,538
8.65%
9,863
7.70%
16,126
4.98%
11,123
6.54%
6,654
7.84%
4,335
5.46%
5,290
5.96%
2,505
5.48%
7,007
5.30%
3,526
5.30%
1,960
6.77%
1,234
5.45%
801
6.56%
385
6.05%
600
4.96%
295
4.00%
161
3.94%
6,895
6.25%
9,789
4.87%
*, (**) significant at the 90% (95%) confidence level
a
θ is not age-specific in this equation
More surprising is the fact that the marginal productivity of the medical input does not
decrease in middle and old age. Beyond the age of 40 the elasticity is more or less constant
for both sexes. This result reflects the significant life-year gains that have been produced in
recent years even at old ages.
The elasticity of the death rate with respect to medical inputs is higher for men than for
women. This holds not only for age groups where women have a higher expenditure due to
8
motherhood but for the entire life-cycle. In old age, the advantage of men reflects their larger
gains in longevity in the study period. The weighted average of θ is 11 percentage points
higher for men compared to women. If one fixes θ across the life-cycle, a 16 percentage point
higher elasticity for men is realized (0.80 as opposed to 0.64). At first glance, given the higher
life-expectancy for women this result is surprising. It can be explained by two factors. Firstly,
the reduction in mortality was larger for men (the gender gap in longevity has been shortened
from 5.77 years in 1997 to 5.09 years in 2003). Secondly, over the entire life-cycle health care
expenditure for women exceeds that for men. The gender difference is even more accentuated
in terms of the marginal costs of life. In the overall estimation with age-constant θ, the
marginal costs of life of women are 30 percent higher than for men, and even more than twice
as high when θ is not age-specific.
The MCL-curve over the life-cycle corresponds with the shape of the value of life, well
known from the literature (see, for instance, Pratt and Zeckhauser, 1996). The willingness to
pay for saving a life decreases beyond a certain age, the driving force being the survival
function which discounts future utility. Murphy and Topel (2005) calculate that the value of a
statistical life at age 80 is 10 percent the value at age 40. My figures regarding the marginal
costs of life are similar and resemble the ones given by Hall and Jones (2005). In view of the
theory given above the com-movement of the value and the marginal costs of life indicates
that the allocation of health spending across the life-cycle is efficient.
The absolute value of the marginal costs of life can not be taken at face value. Firstly, there
are other factors such as the medical progress and the environment which will also affect agespecific death rates. Jones and Hall (2005) assume in their base case that health spending
accounts for only one third of the decline in mortality. If I considered this, the MCL in Table 2
would increase by 50 percent. Moreover, I could take into account that any progress in the
medical sector exceeds the technical progress in the rest of the economy which would
increase the MCL figures even further. Still, the reported figures come close to the survey by
Viscusi and Aldi (2003) who find estimates of the value of life that range from $4 million to
$9 million.
The average annual growth rate in the MCL is 5-6 percent between 1997 and 2003. This
corresponds to the figure in Jones and Hall (2005) who report for the USA in the period 1950
to 2000.
9
4.2 Regional differences
Turning to the regional data, one has to take into account the small population size of some
cantons which can result in zero death numbers for some age groups. For that reason I
combined the data for men and women, extended the age intervals from 5 to 10 years and
aggregated the 26 cantons to 20 regions (for details see column 2 in Table 3). Moreover, at
this disaggregated level death rates still have a large variance so it is not possible to undertake
a robust estimation of the age-specific elasticity parameter. Instead, I estimated a constant θ
while allowing for age-specific efficiency, γ a , and cantonal differences κ k :
log qa ,k ,t = log γ a + log κ k − θ k ⋅ log ma ,k ,t + ε a ,k ,t ,
(10)
The estimation results by OLS, using no instruments for a possible trend in the data, are
presented in Table 3. Only two cantons show a difference in the efficiency level κ k as
compared to Aargau, serving as the benchmark.
A similar result concerns the elasticity of the death rate with respect to health inputs. While
the elasticity is significant in each canton, no significant differences occur between cantons.
Regarding the major regions, the elasticity is somewhat higher in the West and in the NorthWest. However, the differences are remarkably small. Restricting the model to the age groups
30+ renders the differences in the elasticity even smaller. In terms of the parameters, thus, the
health production function does not much differ across the Swiss cantons.
Small differences in θ k imply a large variation in the marginal costs of life as there exist
substantial differences in the level of health care expenditures across cantons. For MCL I find
a range from 3.2 million SFr. in Basle to 11.1 million SFr. in Solothurn. Regarding the
regions, the West has the highest MCL (6.9 million SFr.) while Zurich and the North-West
region show the lowest MCL (5.1 million SFr.).
Regressing MCL to the income per capita in the cantons produces no significant coefficient.
However, in the restricted sample of the age groups 30+ I find a cubic structure: For an
income between 30,000 and 40,000 SFr. MCL increases, between 40,000 and 50,000 SFr. a
flat decrease occurs, followed by a positive gradient for MCL for a higher income. Hence,
there is some support for a positive income effect on the marginal costs of life, noticing that
Costa and Kahn (2004) report an income elasticity of 1.6 for the time period 1940-1980 in the
US.
10
Table 3:
The elasticity of the death rate with respect to health care expenditure (θ),
and the marginal costs of life (MCL), by cantons
κkb
Geneva
Vaud
Valais
Ticino
Mean
Berne, Jura
Midlands
Fribourg
Neuchâtel
Solothurn
Mean
North-West Aargau
Basle, country
Basle, city
Mean
Lucerne
Central
Nid- and Obwalden, Uri
Schwyz, Zug
Mean
Appenzell
East
Glarus, Graubünden
St. Gallen
Schaffhausen
Thurgau
Mean
Zurich
West
** significant at the 95% confidence level,
a
-1.86
-0.52
-0.11
-1.68
-0.18
-0.98
-2.24**
1.81
3.36**b
-1.31**
-3.54
0.95
-0.36
-0.42
-0.33
-0.54
-0.16
0.31
0.13
-0.97
Θ
MCL 2003a
(in 1,000 SFr.)
0.81**
0.65**
0.68**
0.84**
0.74
0.65**
0.75**
0.89**
0.42**
0.62
0.66**
0.81**
1.01**
0.74
0.54**
0.72**
0.71**
0.61**
0.71**
0.71**
0.70**
0.60**
0.64**
0.66
0.74**
8,114
6,429
7,398
4,826
6,924
5,998
4,901
4,684
11,122
6,459
5,650
5,370
3,186
5,123
7,067
5,148
5,432
6,215
4,268
4,939
5,782
6,619
6,132
5,640
5,083
The weighted mean across age-intervals,
b
Growth
rate
(1997-2003)
4.21%
6.47%
5.32%
5.06%
5.15%
9.20%
3.42%
8.85%
7.13%
7.89%
6.72%
6.18%
3.43%
6.21%
7.21%
3.87%
6.90%
6.61%
5.58%
5.09%
8.22%
5.80%
5.08%
6.48%
7.70%
Aargau is the benchmark.
5. Conclusion
Unlike other European countries such as Great Britain, France and Germany, Switzerland has
always followed a decentralized policy in health care. The sickness funds’ premiums are
allowed to differ between regions and subsidies to the stationary sector of health care as well
as health premium subsidies to low-income individuals are governed by the local authorities.
The federalism in health care has led to diverging regional levels of health care provision and
corresponding expenditures (see Crivelli et al., 2006). For instance, the West spends almost
11
twice as much on health care than the East of Switzerland. Interestingly enough, higher
spending comes with a higher longevity: the people in the West live almost one year longer
than the people in the East.
I estimated age-specific production functions relating the death rate to health care
expenditures. Based on the estimated coefficients I then calculated the marginal costs of life at
each age, i.e. the additional expenditure for saving a life. Regarding the regional results, I
found no significant differences for the parameters of the production function, overall
efficiency and the elasticity of the death rate with respect to the health care expenditure. It
thus appears that medical technology is more or less the same across Swiss regions. If one
combines this observation with the existing large differences in the level of health spending,
substantial differences in the marginal costs of life arise. In fact, the regional values of the
marginal costs of life range from 3 to 11 million SFr.
Regressing the marginal costs of life to the income per capita produces some support for the
claim that the value of life is a normal good. The observation of an annual increase of the
marginal costs of life in the study period 1997-2003 of about 4.5 percent, compared to a 1.5
percent growth rate of real per capita income, also vindicates this claim.
Regarding the age-characteristics of medical technology, the elasticity of the death rate with
respect to health spending is higher at young ages. Beyond the age 30, the elasticity is
constant. In particular the elasticity of medical inputs does not decrease in old age, thus
reflecting substantial reductions in mortality in recent years. The marginal costs of life have a
decreasing shape beyond the age of 55, driven by increasing health care expenditure. The
shape of the marginal costs of life across the lifecycles is found to be similar to that of the
value of life well known from the pertinent literature.
The quantitative analysis reveals significant differences in the health production parameters
between men and women. The elasticity of the death rate with respect to health spending is
higher for men. This result even holds in old age. It can be explained by the larger gains in
longevity by men. Lower productivity and higher health care expenditure result in substantial
higher marginal costs of life for women.
Given the gender gap in longevity, Richard Posner argued for transferring the government
spending for health care from old women to old men, because shortening the gender gap
“would give elderly women a greater prospect of male companionship, something many of
them greatly value” (Posner, 1995, p. 277). If the marginal costs of life of women are in fact
larger than that of men, as this study finds, Posner has an even stronger case.
12
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