Journal of Health Economics 50 (2016) 9–26
Contents lists available at ScienceDirect
Journal of Health Economics
j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / e c o n b a s e
Non-monotonic health behaviours – implications for individual
health-related behaviour in a demand-for-health framework
Kristian Bolin a,b,*, Björn Lindgren b,c,d
a
Department of Economics, University of Gothenburg, Gothenburg, Sweden
Centre for Health Economics, University of Gothenburg, Gothenburg, Sweden
Department of Health Sciences, Lund University, Lund, Sweden
d
National Bureau of Economic Research (NBER), Cambridge, MA, USA
b
c
A R T I C L E
I N F O
Article history:
Received 25 March 2014
Received in revised form 2 August 2016
Accepted 16 August 2016
Available online 26 August 2016
JEL classification:
I12
Keywords:
Human capital
Grossman model
Non-monotonic health investments
Health
Steady-state and stable equilibria
A B S T R A C T
A number of behaviours influence health in a non-monotonic way. Physical activity and alcohol consumption, for instance, may be beneficial to one’s health in moderate but detrimental in large quantities.
We develop a demand-for-health framework that incorporates the feature of a physiologically optimal
level. An individual may still choose a physiologically non-optimal level, because of the trade-off in his
or her preferences for health versus other utility-affecting commodities. However, any deviation above
or below the physiologically optimal level will be punished with respect to health. Distinguishing between
two individual types we study (a) the qualitative properties of optimal time-paths of health capital and
health-related behaviour, (b) the perturbations of the optimal time-paths that result from changes in
exogenous parameters, and (c) steady state properties. Predictions of the model and the implications for
empirical analysis are discussed at length. Some comments on potential future extensions conclude the
paper.
© 2016 Published by Elsevier B.V.
Most human behaviours are related to health. Individual health
affects consumption patterns, but consumption patterns also affect
individual health. While the intention of health-care utilisation is
either to improve current health, whenever it has fallen below a
certain illness-defining threshold value, or to prevent future illness,
rather than to consume it for the sake of its direct utility (Arrow,
1963, p. 948), the intention of many other behaviours may be
twofold: both to gain direct consumption utility and to improve
health (or to decrease the risk of illness). The latter category includes, for instance, physical exercise, certain consumption and
composition of food, alcohol consumption and, as a matter of fact,
any recreational activity (art, literature, music, etc). Obviously, health
effects may be more or less unintentional, and certain consumption patterns may also be detrimental to your health. Smoking is
an unambiguous example of the latter.
Smoking is always bad for your health – and increasingly so with
increased consumption (Colditz, 2000; Doll and Peto, 1976; Doll et al.,
1994; Vineis et al., 2004).1 In contrast, there appears to be a physiologically determined, individually optimal level of activity (greater
than zero) as regards, for instance, physical exercise, food intake,
alcohol consumption, and sleep, implying that activity levels below
or above that level would result in a level of health which is lower
than the maximum achievable level; see Fig. 1 for an illustrative
example of a typically U-shaped relation.
A consistently positive association between physical-activity level
and health-related quality of life has been found (Bize et al., 2007).
Certainly, too small amounts of physical exercise means that the
human body atrophies and that the risks of several diseases, including coronary heart disease, hypertension, stroke, diabetes,
depression, osteoporosis, and cancers of the breast and colon, increase (Garrett et al., 2004). However, too much or too intensive
physical exercise means that the human body will wear down and/
or that the risk of injury increases (Howatson and van Someren,
2008; Ji, 2001; Locke, 1999; Morton et al., 2009; Randolph, 2008;
Tisi and Shearman, 1998). A U-shaped association between allcause mortality and dose of jogging has been found (Lee et al., 2015;
* Corresponding author at: Department of Economics, University of Gothenburg,
Gothenburg, Sweden.
E-mail address: [email protected] (K. Bolin).
1
There is, however, recent evidence showing that nicotine at low to moderate levels
can have health and therapeutic benefits; specifically, the ability to act as a
neuroprotectant has been demonstrated (Hurley et al., 2012).
1. Introduction
http://dx.doi.org/10.1016/j.jhealeco.2016.08.003
0167-6296/© 2016 Published by Elsevier B.V.
10
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Fig. 1. Relative risk of mortality in relation to alcohol intake for middle-aged (-) and elderly (…..) people. Risk was set at 1.00 at lowest mortality at 1–6 beverages per
week. Vertical lines are 95% confidence intervals for point depicting estimates for all subjects. (Reproduced from Grönbaeck et al., 1998 with permission from the Oxford
University Press).
Schnohr et al., 2015). A varied and balanced diet is emphasised in
guidelines on healthy eating; see, for instance Swedish National Food
Agency (2015). Too little food or too one-sided diet leads to health
problems (Steinhausen, 2002). Too much also creates health problems (Steinhausen and Weber, 2009), in particular in combination
with too little physical exercise.
Overweight (25 ≤ BMI < 30)2 and obesity (BMI ≥ 30) increase the
risks of asthma, coronary heart disease, hypertension, diabetes, osteoarthritis, and cancer, including cancers of the breast and colon
(Colditz, 1999; Must et al., 1999; Dal Grande et al., 2009). Also underweight (BMI < 18.5) has been shown to be associated with health
problems; for instance, coronary heart disease, diabetes, and gallbladder disease (Must et al., 1999). Light to moderate drinkers are
at lower risk of coronary heart disease, stroke, diabetes, and gallstone disease than non-drinkers, while an increasing intake increases
the risks of dementia, liver cirrhosis, pancreatitis, osteoporosis, and
most cancers, including cancer of the oesophagus, breast, pancreas, colon, and rectum (Grönbaeck, 2009). Finally, both short and long
sleep durations appear to be related to increased likelihood of obesity,
2
BMI (body mass index) is calculated as weight in kilograms divided by the square
of the height in meters (kg/m2).
diabetes, hypertension, and cardiovascular disease (Buxton and
Marcelli, 2010; Sabanayagam and Shankar, 2010).
It should be emphasised, though, that which level of physical
exercise, food intake, alcohol consumption, and sleep is physiologically optimal differs among individuals, and if you have good genes
and/or are lucky, you may suffer less from “unhealthy” behaviour
than less advantaged people.
In general terms, these associations have been known for decades.
Yet, there are no clear temporal trends worldwide towards healthier
life-styles (Knuth and Hallai, 2009), and the population variance of
these behaviours is large; for instance, many people do not perform
any, or very little, physical exercise, others perform very large amounts.
Some of this variance may be readily understood within the demandfor-health model (Grossman, 1972a, 1972b) both in its original version
– see Muurinen and Le Grand (1985) and/or with minor extensions
– see Galama and van Kippersluis (2013). The observed variance,
however, seems to be greater than what would be expected, solely
taken the variability in physiologically determined, individually optimal
level of activity into account (Cutler and Gleaser, 2005).
In this paper, we develop a version of Grossman’s demand-forhealth model, assuming that there is a (strictly positive)
physiologically optimal level of health behaviour and that the individual will be punished with lower health if exerting too much
or too little of that behaviour. The emphasis is on extending theory
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
– to construct a model, which can improve the understanding why
individuals engage in behaviours that are sub-optimal with regard
to health. Such an analysis also has important implications for designing and evaluating health policies. Policy-relevant conclusions,
though, will depend very much on to what extent theoretical predictions will be confirmed and associations estimated in empirical
analyses. Thus, predictions of the model and the implications for
empirical research will be discussed at length.
The structure of the paper is as follows. Next, we will introduce basic features of our model as related to previous theoretical
research. Thereafter, we will present our model formally and derive
optimality conditions. Following this, we will characterise the properties of the dynamic system that we use in terms of stability
conditions and admissible time paths of behaviour and the health
stock. Differences in initial health and in the sensitivity to deviations from the health-behaviour activity level at which there is no
punishment in terms of reduced health will be chosen for further
analyses. In the concluding discussion, the possibilities for empirical analysis will be discussed, using available data in Sweden (and
other Nordic countries) as examples. Comments on potential future
extensions of the model will end the paper. An Appendix and an
on-line supplement provide technical details.
2. Our model in relation to previous theoretical developments
The demand-for-health model extended the human capital theory
by explicitly incorporating health and recognising that there are both
consumption and investment motives for investing in health
(Grossman, 1972a, 1972b). It resulted in an economic theory of individual health-related behaviour. The basic features of the model
are (1) that the individual demands health (a) for its utility enhancing effects (the consumption motive), and (b) for its effect on the
amount of healthy time (the investments motive), (2) that the demand
for investments in health is derived from the more fundamental
demand for health, (3) that the investments in health are produced
by the individual, and (4) that the stock of health depreciates at each
point in time. Thus, investments in health require the active or passive
presence of the individual, and individuals may differ, for instance,
because of genetics or education, to what extent a specific behaviour,
for instance, physical exercise, would lead to an increase in health.
The demand-for-health model, as presented in Grossman’s original paper, relied by necessity on a number of simplifying
assumptions, “all of which should be relaxed in future work”
(Grossman, 1972a, p. 247). Certainly, the model has been extended, but not towards some general economic theory of health
behaviour. Rather, the extensions have been made in many different directions, relaxing some of Grossman’s assumptions at a time
but often introducing new simplifications. The objective has been
to let the focus be on one specific phenomenon and to provide clear
and testable model predictions relating to this phenomenon. This
is a way of theorising that has a long tradition; it stems at least back
to the English philosopher William of Ockham (c. 1287–1347) and
his famous Razor: “Don’t multiply entities beyond necessity.”
Thus, while Grossman’s original model assumed complete certainty of life and death, several authors have incorporated uncertainty
and insurance (Asano and Shibata, 2011; Chang, 1996; Cropper, 1977;
Dardanoni and Wagstaff, 1987, 1990; Ehrlich, 2000, 2001; Liljas, 1998,
2000; Picone et al., 1998; Selden, 1993). Predictions are, inter alia,
that risk-averse expected-utility maximising individuals would make
larger investments in health and have larger expected health stocks
in the presence of uncertainty than they would without it; insurance, especially social insurance, would lower the optimal stock of
health. Predictions are in accordance with Grossman’s discussion
of the effects of introducing uncertainty (Grossman, 1972a, pp. 19–
21). Grossman believed that the easiest way to introduce uncertainty
would be by changing the character of the depreciation rate from
11
a deterministic to a probabilistic one. Generally, the way uncertainty has been introduced is as some random element of the stock
of health. This is one of the three uncertainties that Arrow recognised
in his seminal paper, “Uncertainty and the welfare economics of
medical care” (Arrow, 1963). The other two – uncertainty in the diagnosis of illness and uncertainty in the outcome of treatment –
remain to be analysed. The latter is obviously not confined to medical
treatment only but is involved in all kinds of investments in health.
Thus, Dowie (1975) suggested a portfolio approach to health
behaviour, the simple basic message of which is that the individual should diversify, when making investments that affect health.
Muurinen (1982) introduced a “use-related” deterioration of health
capital influenced by, inter alia, education, and a health-investment
efficiency parameter declining with age, replacing the household production framework used by Grossman. Liljas (1998) assumed the rate
of depreciation of health capital being a negative function of the level
of health and, hence, becoming endogenous instead of exogenous.
Liljas (2000, 2002) examined cases where financial markets are imperfect rather than perfect. The allocation of health and health
investments within families rather than of a single individual has been
analysed for various family structures and objective functions –
Jacobson (2000) for the case when the family has a common utility
function; Bolin et al. (2001) for the case when spouses are Nashbargainers and Bolin et al. (2002b) when they act strategically; and
Bolin et al. (2002c) for the case when an employer has incentives for
investing in the health of one of the family members. Social interaction with friends and relatives and the importance of social capital
were analysed by Bolin et al. (2003). The impact of replacing Grossman’s assumption of constant returns to scale in the production of
health investment by decreasing returns was analysed by Ehrlich and
Chuma (1990) and Galama (2011), showing, inter alia, that this creates
incentives for the individual to postpone some of a planned health
investment to later time periods. Galama and Kapteyn (2011) and
Galama et al. (2013) made the original model more general by allowing for corner solutions. Attempts are being made to relax the
assumption of an exogenous stock of educational capital and provide
a model in which health and educational capital are determined simultaneously (Galama and van Kippersluis, 2013; Carbone and
Kverndokk, 2014).
In the model that we present here, a single individual maximises
utility under complete certainty over some planning horizon; it might
be convenient to think of an adult with a specific initial level of health,
who makes plans for the rest of his or her life. The individual’s healthrelated behaviour not only affects the individual’s production of health
but also provides instantaneous utility. While our model differs from
Grossman (1972b), the phenomenon per se was treated by Grossman
(1972a, pp. 74–79) as joint production of “commodities”; market goods
and services that are demanded as inputs into the production of commodities other than health might also be positive or negative inputs
into the health investment production function. Furthermore, Forster
(2001) studied health-related effects of there being two types of consumption: good for health and bad for health. However, both Grossman
(1972a) and Forster (2001) assumed that the effect on health was
monotonic in consumption, either positive or negative. In our case,
the relationship between the amount of behaviour and its health effect
is assumed to be non-monotonic (i.e., positive for some range and
negative for some other range of behaviour), at the same time as the
positive utility effect is monotonic. Our model belongs to the same
family as the ones developed by Becker and Murphy (1988) and by
Dockner and Feichtinger (1993), i.e. models characterised by a
behaviour yielding instantaneous utility as well as adding to a stock
that influences future optimal choices.3
3
We are grateful to one of the anonymous referees for making us aware of the
similarities.
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K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Box 1Denotation of variables.
H: Health capital
Hmin: Smallest possible health stock (“death” stock)
H0: Initial health stock
B: Health-related behaviour
B*: Health-related behaviour that gives no punishment
Bmax: Maximal affordable health-related behaviour
δ: Rate of depreciation of the health stock
ρ: Rate of time preferences
φ: The rate at which gross health investments are offset, when the activity level deviates from B*
I: Gross health investments
Z: Health-unrelated consumption
p: Price of Z
q: The composite price of market goods and services used in the production of B
w: Wage rate
π(q, w): Unit cost of B
y: Income
Ω: Total time
τ: Sick time
U(H, B, Z): Instantaneous utility
u(H): Simplified instantaneous utility of health
v(B): Simplified instantaneous utility of B
k: Simplified marginal utility of Z
λ: Shadow price of health capital
T: Length of the life cycle
t: Time
Some simplifying assumptions were made in order to facilitate
the derivations of results of specific interest for the purpose of the
present study. We avoid an explicit treatment of the individual’s timeallocation problem. Instead, we utilise the individual’s cost function,
pertaining to the allocation problem that he or she faces. This means
that we implicitly assume that the individual has solved the timeallocation problem. Furthermore, like Liljas (1998) and Forster (2001),
we assume that there are no financial markets; the only way to transfer resources over time would be to invest in health. We also assume
that preferences, prices and depreciation rate are all time invariant. To simplify further, we abstain from any direct treatment of the
possible effects of education, even though several links between education and non-monotonic health behaviour might be plausible.4
Our overall objective is to develop a theoretical framework that
improves and expands the understanding of individual healthrelated behaviours that influence health in a ∩ -shaped way. More
specifically, our aim is to provide a refined theoretical basis for empirical analyses of individual health-related behaviour and a tool
for analyses of policy interventions that target individual healthrelated behaviours. Here, we will focus on developing the model
and on demonstrating its usefulness for understanding the relationship between individual-specific (or behavioural-specific)
sensitivities of deviating from what would be physiologically optimal
and what is actual individual behaviour.
3. Formal theoretical model
In this section, we present our formal model – assumptions about
technology, budget constraint, and preferences. We further define
the individual’s physiologically optimal level of health behaviour
and distinguish between two types of behaviour – the overexerting and the under-exerting, compared to the physiologically
optimal level. The denotations used are summarised in Box 1.
We consider a version of the demand-for-health model in continuous time, in which health at each point in time, H(t), is produced
4
A more educated individual may, for instance, be more aware of the detrimental effects of deviating from the physiologically optimal activity level.
by the individual through a specific health behaviour, B(t), which
also provides instantaneous utility. The influence on health is nonmonotonic and single-peaked, while the effect on utility is
monotonically increasing. The individual also derives utility (a) from
the health stock, and (b) from health-unrelated consumption, Z(t).
3.1. Technology, budget constraint and preferences
3.1.1. Technology
The health-related behaviour, B(t), is produced by the individual using market earnings and own time. For simplicity, we assume
that the technology employed is homogenous and exhibits constant returns to scale. Formally, the cost function is5:
C ( B (t )) = π (q, w ) B (t ),
(1)
where π(q, w) is the one-unit cost of producing the behaviour, q is
the composite price of market goods and services used in the production, and w is the wage rate
3.1.2. Budget constraint
The consumption good is bought on the market at a constant
price, p. There are no financial markets and, hence, total spending
at time t equals market income at time t. Therefore, the individual’s budget constraint is:
pZ (t ) + π (q, w ) B (t ) = y ( H (t )),
(2)
where y denotes full market income (the market value of total time),
and p the price of the consumption good. Health determines the
total amount of time available for the individual to allocate between
its two productive uses, behaviour B and market work. Thus, full
market income is a function of health and is given by:
y (t ) = w (Ω − τ ( H (t ))),
(3)
5 The technology is formally represented by a production function that is homogenous of degree 1 in the quantity of B(t), and the dual cost-of-behaviour function
is homogenous of degree 1 with respect to the quantity of B(t), i.e., the marginal
cost of producing the behaviour is constant.
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
where Ω is total time and τ ( H (t )) is time spent being sick. Formally, the amount of time spent being sick is inversely related to the
stock of health capital, i.e., τH < 0.6 Further, we assume that the productivity of health in producing healthy time is diminishing, i.e.,
τHH > 0. Thus, potential income is increasing and concave in health
capital, since yH (t ) = − wτ H > 0 ; and yHH (t ) = − wτ HH < 0 .
3.1.3. Preferences
Individual preferences are represented by the instantaneous utility
function, U ( H (t ) , B (t ) , Z (t )), that is (a) monotonically increasing and
twice differentiable in all arguments, (b) time additive, and (c) jointly
concave in ( H (t ) , B (t )) . We also assume that U HB > 0 .
3.2. The physiologically optimal level of health behaviour
Here we define the physiologically optimal level of health
behaviour and two different types of individuals distinguished by
their behaviour – those who exert too much and those who exert
too little of the health behaviour in relation to his or her physiologically optimal level. The distinction is fundamental for our
following analysis. For the phenomenon to occur at all, maximising
utility must result in a solution different from a behaviour, which
maximises health. In Section 4, the individual’s control problem is
presented, equilibrium conditions are derived and their relation to
the two types of individuals are explained. We start by stating the
equation of motion for the stock of health capital.
3.2.1. Motion of health capital
The positive health effect produced by the behaviour, B(t), is partially offset by natural depreciation – at rate δ(0 < δ < 1) – of the existing
stock of health capital. For tractability of dynamic analysis, we consider a model in which the rate of depreciation is time-independent.7
Further, the behaviour, B(t), transforms into gross health investments, I(t), according to a non-monotonic single-peaked relationship.
Thus, the equation of motion for the stock of health capital is:
2
H (t ) = B (t ) − ϕ ( B (t ) − B*) − δ H (t ) ≡ I (t ) − δ H (t ).
(4)
Gross health investments are given by the first two terms. The
third term is depreciation of the health stock, at rate δ, at each point
in time. The behaviour B(t) has simultaneously both a beneficial and
an offsetting adverse effect on gross health investments, given by
B(t) and ϕ ( B (t ) − B*)2 , respectively. Thus, the parameter φ (φ > 0) indicates the rate at which gross health investments are offset, when
the activity level deviates from B*. It reflects differences between
behaviours as well as differences between individuals. That is, φ
varies between individuals, for the same type of behaviour, and
between behaviours for the same individual.
3.2.2. The physiologically optimal level
Starting at B (t ) = 0 and moving towards more B-behaviour
will gradually reduce the adverse effect of deviating from B * .
The adverse effect exactly balances the positive effect at
B (t ) =
1
1 + 4 B *ϕ 8
+ B* −
. At the activity level B (t ) = B* , the offset2ϕ
4ϕ 2
6 Throughout the paper, a subscript indicates a partial derivative. Following established practice, though, we use the dot notation for the time derivative.
7
We follow Forster (2001) who employed a model with a time-invariant rate of
depreciation, ensuring that the (current value) solutions to the model are autonomous and, hence, tractable for diagrammatic analysis (Forster, 2001, pp. 614 and
622). Grossman (1972a, 1972b), Muurinen (1982), Liljas (1998), Jacobson (2000),
Bolin et al. (2001, 2002b, 2002c), Galama (2011) and Galama et al. (2013) examined models with time-dependent rates of depreciation.
8 Thus, the produced amount of gross health investments might be negative. In
the original demand-for-health model, it was assumed that the amount of gross health
investments is positive.
13
ting effect is minimised, and the marginal contribution to gross health
investments is positive. The largest amount of gross health invest1
1
+ B* , at B (t ) =
+ B* .
ments that can be produced is I max (t ) =
4 ⋅ϕ
2⋅ϕ
This will be referred to as the physiologically optimal level of the
B-behaviour. Increasing B beyond that point would start to reduce
1
1 + 4 B *ϕ
, the
+ B* +
2ϕ
4ϕ 2
adverse and the beneficial effects are (again) equal. Notice that a
larger φ means that the activity level that will maximise gross health
investments approaches B*.
the health effect of the behaviour. At (t ) =
3.2.3. Two types of individual health behaviour: the under-exerting
and the over-exerting type
We distinguish between two individual types according to the
individual’s choice of B: those who exert less than the physiologically optimal level (the under-exerting type) and those who exert
more (the over-exerting type).9 A third type might be possible to define
– those who exert exactly the physiologically optimal level of
B-behaviour – but this will be considered as an exception or a borderline case to which we will refer only for comparisons.10 Fig. A1
(Appendix) illustrates the influence on gross health investment produced by the B-behaviour.
4. The utility-maximising health-related behaviour
In this section we deduce which factors may determine the type
of individual. In order to do so, we first present the individual’s
control problem and derive the equilibrium conditions.
4.1. The individual’s control problem
The individual’s objective is to maximise his or her present and
future utility over a finite time horizon. Formally, this means solving
the inter-temporal optimisation problem of choosing the time path
of health capital that maximises utility over time by engaging optimally in the health-related behaviour. We assume a fixed planning
horizon, T, and a fixed end-point, H (T ) = H min , where H min is the
“death” stock. Thus, discounting future utility at the rate ρ, the individual acts as if solving the following problem11:
T
max B(t ) ∫ e − ρ⋅tU ( H (t ), B (t ), Z (t )) dt
0
2
subject to: H (t ) = B (t ) − ϕ ( B (t ) − B*) − δ H (t ),
pZ (t ) + π (q, w ) B (t ) = y ( H (t )),
H (0) = H 0, H (T ) = H min, H (t ) > H min ∀t ∈[0, T ].
(5)
9
The under-exerting type might be seen as a case in a somewhat extended realm
of the standard demand-for-health model except for the fact that we have introduced a limit to the extent to which health-related behaviour can be used to reduce
the decline in health. The similarity is obvious, if we let φ = 0. The over-exerting type,
on the other hand, behaves similarly to the hazardous-goods consuming individual in Ippolito’s (1981) model. The difference is that consumption affects survival in
Ippolito’s model, while it reduces the health stock in ours. We are grateful to one
of the anonymous reviewers for making us aware of these similarities.
10
We are indebted to one of the anonymous reviewers for drawing our attention
to the fact that the physiologically optimal-exerting type is nothing but an idiosyncrasy – both in theory and in the real world. For theory, see further footnote 16.
11
Our specification of the individual’s optimisation problem as a fixed end-point
problem means that we do not have to examine the shadow price of health capital
at t = T (Chiang, 1992, p. 182). In order to analyse the optimal length of life, the problem
can be formulated as a horizontal terminal line problem in which the terminal time,
T, is free, and, hence, that the terminal state is restricted to H (T ) ≥ H min ; see, for instance, Ehrlich and Chuma (1990).
14
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Over-exerting type
”Optimally” exerting type
Under-exerting type
Fig. 2. The relationship between φ and the utility-maximising level of B(t) for given preferences but different φ’s. The curve represents the marginal instantaneous utility
of B, while the straight lines illustrate marginal health benefits (cost) of B. The optimally exerting type is used as a reference case, and is represented by the solid line; the
line crosses the marginal health benefit curve at B* +
1
1
. The upper (lower) dotted line illustrates an over (under) exerting type, since the optimal level of B (t ) > (< ) B* +
.
2ϕ
2ϕ
4.2. First-order conditions
The solution to the maximisation problem is obtained by means
of the maximum principle of optimal control theory, which gives
necessary and sufficient conditions for the optimal control of B(t),
given the time path of the state variable H(t).12 Substituting for Z(t)
using the budget constraint (2), the current-value Hamilton function for the maximisation problem becomes:
y ( H (t )) − π B (t ) ⎞
⎛
H (t ) = U ⎜ H (t ) , B (t ) ,
⎟⎠
⎝
p
(
)
(6)
+ λ (t ) B (t ) − ϕ ( B (t ) − B*) − δ H (t ) ,
2
where λ(t) ≥ 0 is the current value costate variable (the marginal
value of health capital).13 In what follows, the time argument (t) is
omitted where convenient. The maximum principle yields the following equation of motion for the costate variable, λ:
w
λ = − HH + λρ = −U H + U z τ H + λ (δ + ρ ).
p
(7)
Further, for all points in time, the control variable, B(t), must be
chosen in order to satisfy the first-order (sufficient) condition14:
HB = U B − U z
π
+ λ (1 − 2ϕ ( B − B*)) = 0.
p
(8)
Condition (8) implies that B(t) is chosen so that the sum of (a)
the instantaneous marginal utility of B(t), (b) the instantaneous mar-
12 Textbook introductions to optimal control have been published by, e.g., Leonard
and Van Long (1992) and Shone (2002). Shone (2002) is more elementary and also
gives an introduction to numerical methods for solving dynamic optimisation problems. Caputo (2005) provides a more advanced treatment of optimal control theory.
13 The rationale for assuming that λ (t ) ≥ 0∀t ∈ [0, T ] , is that U − U w τ > 0
H
Z
H
p
∀ t∈[0,T].
14
By the Mangasarian sufficiency theorem, Equations (7) and (8) are necessary and
sufficient conditions for an optimum, if the Hamiltonian function, H , is jointly and
strictly concave in (H , B ) , and if λ(t) ≥ 0 ∀t. Strict and joint concavity of the Hamiltonian function is guaranteed here, since both U(·) and the equation of motion of
health capital (Equation 2) are strictly and jointly concave in (H , B ) .
ginal cost of B(t), in terms of the marginal loss of utility induced
by the necessary reduction of Z(t) required in order to increase B(t)
by one unit, and (c) the marginal health cost (or benefit) produced by B(t) – the last term – equals zero. We restrict our attention
to cases in which the individual does not spend all resources on B(t).15
4.2.1. Utility-maximising health-related behaviour and type of
individual
The analysis of utility-maximising choices of B(t) is facilitated
by a reformulated version of Equation (8):
UB − Uz
1 ⎞⎞
π
⎛
⎛
= λ 2ϕ ⎜ B − ⎜ B* +
.
⎝
⎝
p
2ϕ ⎟⎠ ⎟⎠
(8’)
The left-hand side is the instantaneous (net) marginal utility and
the right-hand side is the marginal health cost of the B-behaviour.
The optimal level of the B-behaviour is found, when the marginal
utility of the behaviour equals its marginal cost in terms of its effects
on the health stock. This is illustrated in Fig. 2 for different φ’s.
Prediction 1. Behaviours that influence health non-monotonically are
exerted at levels that are either lower or higher than what would be
physiologically optimal.
The over-exerting type experiences a positive marginal cost of
the B-behaviour, while the under-exerting type faces a negative marginal cost (a benefit), since an increase in B would increase the health
benefit. The individual would be an under-exerting type, if the intersection between the marginal utility and the marginal cost curves
1
, and
is to the left of the physiologically optimal level, B (t ) = B* +
2ϕ
an over-exerting type, if it is to the right.16 In Fig. 2, the individual
facing the higher rate at which deviations from the physiologically optimal level of B(t) are punished, φo, has a physiologically optimal
That is, π ⋅ B (t ) < y (H (t )) ∀t .
1
The existence of an optimally-exerting type would require that B = B* +
and,
2ϕ
π
hence, that U B = U Z .
p
15
16
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
level of B (t ) = B* +
1
, and is classified as an over-exerting type
2ϕ o
individual.17
The decision to behave at levels below or above the physiologically optimal level is determined by preferences, the sensitivity of deviating
from the optimal level, and prices.
Prediction 2. The stronger the preferences for the health-behaviour
activity, the lower its unit cost and the higher the sensitivity to deviations from the level at which there is no punishment in terms of reduced
health, the more likely the individual would over-exert the activity, i.e.,
utilise it more than what is physiologically optimal.
Individual characteristics and exogenous parameters that determine the intersection between the marginal utility curve and the
marginal cost curve determine behaviour and, hence, may also be
type determinants. First, the location of the marginal utility curve
is determined by the preferences for B(t), and Z(t), the one-unit cost
of B(t), π, the price of Z(t), p, and the health stock, H(t). The stronger the preferences for B(t), the more likely it is that the individual
would be an over-exerting type (stronger preferences for B(t) imply
a higher location of the marginal utility curve). Similarly, the lower
the one-unit cost of B the more likely it is that the individual would
be an over-exerting type.18 However, neither the price level of Z(t)
nor the health stock has a clear-cut effect on the likelihood that the
individual would be an over-exerting type. Second, the location of
the marginal cost curve is determined by the shadow price of health
capital, λ(t), and by the rate at which deviations from B* adversely
influence gross health investments, φ. The relationship between φ
and the likelihood that the individual is an over-exerting (or underexerting) type was illustrated above. Thus, all these factors, except
the shadow price of health capital, are potentially type determinants.
To see why the shadow price cannot be a type determinant notice
that, for given values of B* and φ, the marginal cost curve will in1
. This means that any shift
tersect the horizontal axis at B (t ) = B* +
2ϕ
in the curve, due to a change in λ(t), must be a rotation around this
point.19 Although a rotation of the marginal cost curve will change
the optimal level of B(t), it cannot alter the type classification of the
individual. The reason is that the horizontal position of the intersection between the curves will only approach, but never reach, the
physiologically optimal level of B(t) when the shadow price of health
capital increases (a higher λ makes the marginal cost curve more
steep). The rate of depreciation of the health stock, δ, and the rate
of time preferences, ρ, influence behaviour through the shadow price
of health capital only and, hence, cannot be type determinants.20
5. The dynamics of health-related behaviour and health
In this section, we will, first, illustrate typical shapes of the utilitymaximising time-paths of behaviour and health for an individual
endowed with a large health stock and for an individual with a small
health stock, respectively, employing state phase analysis. This will
17 Fig. 2 can be interpreted either as the same individual facing two different
behaviours (different φ’s) or two different individuals facing the same behaviour but
different sensitivities to deviations from the physiologically optimal level of B(t).
18
The shift of the marginal utility curve induced by a change in π is:
−U BZ
19
B
πB
1
+ U ZZ 2 − U Z < 0 .
p
p
p
The slope of the marginal health cost curve is λ2φ, and its intercept with the
1⎞
⎛
vertical axis is −λ 2ϕ ⎜ B* +
. Thus, an increase in λ increases the slope of the curve,
⎝
2ϕ ⎟⎠
and shifts its intersection with the vertical axis downward.
20 This is obviously true for a given amount of health capital, i.e., at t = 0. In Section
5, we demonstrate that for given exogenous parameters, the individual cannot shift
between types.
15
be followed by an analysis of the effect on the time paths of a shift
in the exogenous parameter φ (comparative dynamics). We will focus
on the sensitivity of deviations from B*, since this parameter captures the core and the distinguishing feature of our framework –
the non-monotonic influence on health of B-behaviour.
The utility function employed hitherto allows for a range of
dynamic behaviours. In what follows, we will restrict our attention to utility functions that are additively separable and quasilinear, and demonstrate that this implies one specific type of
dynamics (a specific phase portrait). Thus, we will employ the following utility function: U ( H (t ) , B (t ) , Z (t )) = u ( H (t )) + v ( B (t )) + kZ (t );
the assumption of strict concavity (jointly) in B, H is maintained.21,22
For readability, the presentation in the following sections is concise
with most mathematical derivations performed in the Appendix and
the online supplement.
5.1. The phase portrait and the significance of initial health
Fig. 3 illustrates the lifetime dynamics of our model. The phase
portrait shows (a) the principal shapes of the stationary loci, assuming that the upper branch of the B (t ) = 0 locus is convex and the lower
is concave, and (b) the direction of admissible trajectories (those
which meet the optimality conditions corresponding to problem (5)).23
The admissible set is bounded by H0, Hmin and Bmax, where Bmax is
the maximum affordable level of the B-behaviour given the level of
health.24 The stable branches for the under-exerting type as well as
examples of admissible and non-admissible trajectories are drawn
as solid arrows and dashed-dotted lines. A solution to our optimisation
problem (5) corresponds to one particular admissible trajectory, which
starts at H0 and ends at Hmin.25 Obviously, the levels of initial and endpoint health determine the admissible set. We will focus on the case
in which initial health is below the maximum level of health that
could be sustained if the B-behaviour was chosen at the physiologically optimal level.26 Examples of admissible trajectories are A and
B, which both satisfy the optimality and boundary conditions and,
hence, are solutions to problem (5), albeit with different planning horizons (different Ts), for the under-exerting type individual. Similarly,
trajectory C illustrates a possible solution to the optimisation problem
for the over-exerting type individual. Trajectories D and E are examples of admissible trajectories that do not meet the boundary
condition and, hence, cannot be solutions to problem (5).
21
An inspection of the Jacobian matrix for the H , B system shows that the local
stability properties of steady state cannot be determined for our general mathematical formulation, which means that steady-state comparative statics and dynamics
analyses in general produce inconclusive result.
22
Quasi-linear utility functions have been extensively applied in economic analyses of the family, and related issues; see, for instance, Chang (2007, 2009); Chang
and Weisman (2005); Konrad and Lommerud (2000); Konrad et al. (2002).
23
The curvature of each branch of the B (t ) = 0 locus cannot be deduced without
further assumptions about functional forms. In our illustrations, the upper branch
is convex and the lower is concave, which would be the case if the third-order derivatives of the utility function and τ are zero (see the online supplement).
24 From the budget constraint we obtain, setting Z (t ) = 0 : π ⋅ B max = y ( H (t )) and,
hence, it follows that the Bmax -locus is increasing and concave in health. Further,
for both the under-exerting and the over-exerting types, there is a smallest and a
largest level of B that maximised instantaneous utility. These levels will never be
attained, since the marginal value of health capital is always positive.
25 As suggested by the trajectories in Fig. 3, a steady state is saddle point stable
for both the under- and the over-exerting types (a formal proof is given in the
Appendix). Comparative statics results, of the steady state, are provided in the Appendix (the principle for the calculations) and the online supplement (for specific
parameters).
26 In the case when both end-point and initial health are above the maximum sustainable health all admissible trajectories would be in the far right part of the phase
diagram in Fig. A2.
16
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
C
Over-exerting type
E
D
Under-exerting type
B
A
Fig. 3. Illustration of stationary loci, admissible sets and trajectories for the control- and state variable. Equilibrium points are marked with a dot. The parabola shows the
max
H (t ) = 0 locus, the hyperbolas the B (t ) = 0 locus. The dotted curve, B , marks the maximum affordable level of B for each amount of health capital. Trajectories A, B and
C fulfil the end-point condition, H (T ) = H T , and, hence, are possible solutions to the optimisation problem. Trajectories D and E, however, do not satisfy the end-point condition, and are therefore not a possible solution to the optimisation problem.
Prediction 3. For given individual characteristics and prices, the individual will not change his or her type from over-exerting to underexerting or vice versa during his or her life time.
Notice that if all exogenous parameters of the model remain constant during the entire planning period, an individual cannot behave
first as an under-exerting type individual and then as an overexerting type individual (or vice versa), since an admissible trajectory
1
cannot cross the horizontal line at
+ B* . The reason is that, for
2⋅ ϕ
the under-exerting type, the instantaneous net marginal utility of
the B-behaviour is (strictly) negative, and decreasing, in B(t) and,
hence, the utility-maximising level must always be strictly below
the physiologically optimal level. An analogous argument applies
to the over-exerting type.
Prediction 4. Individuals endowed with relatively high health levels
may initially behave more healthy but eventually behave less healthy
over time, while individuals endowed with low health levels always
behave less healthy over time.
The components of the phase diagram – stationary loci and admissible trajectories – are invariant with respect to the health
endowment and the “death level” of health. This allows us to illustrate, in the same phase diagram, the effects on utility-maximising
time paths of having a high initial health stock compared to starting with poor health, and how health endowments influence the
two types of individuals. Fig. A2 (Appendix) illustrates admissible
time paths originating from a high and a low level of initial health,
respectively.
Consider the under-exerting type and a high level of initial
health, H0 (the analysis pertaining to the over-exerting type is
analogous). Two typical time paths can be discerned, A and B,
involving either a decreasing or an increasing-decreasing time
path of the B-behaviour, and, in both cases, a decreasing health
stock. Assume that there are two under-exerting type individuals
who differ only as regards the preferences for the B-behaviour.
The individual following the B trajectory has relative strong preferences for the B-behaviour and, hence, will choose a higher level
of B(t) during a first part of the life cycle. Thus, in order to reach
the end-point condition level of health, Hmin, at t = T, the individual following the B trajectory must exert lower levels of B during a
last part of the life cycle.
Next, consider the same two under-exerting individuals being
endowed with a low level of initial health, H10 . In this case, both individuals will follow a decreasing time path of the B-behaviour, but
the individual with relative strong preferences for B will initially
increase his or her health stock according to trajectory C, while the
individual who prefers lower levels of B will typically follow a trajectory such as D. To sum up, the under-exerting type individual,
who is initially relatively healthy, will follow a trajectory with decreasing health, since his or her preferences for the B-behaviour are
not strong enough to offset the depreciation of the health stock.
However, the relatively healthy individual may increase his or her
B-behaviour for some initial part of the life cycle. In contrast, a relatively unhealthy individual may increase his or her health stock at
first, since preferences for B may be sufficiently strong to induce
health investment that more than offset the depreciation of the health
stock.
The results are symmetrical between under- and over-exerting
individuals: The over-exerting individual endowed with a relatively high level of health and has relatively weak preferences for B, may
decrease his or her B-behaviour for some initial part of the life cycle
(trajectory E), and the (over-exerting) individual endowed with a
low level of health and has relatively weak preferences for the
B-behaviour will initially increase his or her health stock (trajectory F). This is illustrated in upper part of Fig. A2.
Further, it is clear from Fig. A2 that the range of admissible
choices of the B-behaviour which are compatible with the endpoint condition is determined by the initial endowment of health,
since B(0) must be below the stable branches (above for the
over-exerting type).
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
17
Fig. 4. Illustration of perturbations of the time paths of (H, λ ) that results from a change in φ. Under- (B < B*) and over-exerting types, the H ϕ = 0 and the λϕ = 0 loci divide
the phase plane in 4 quadrants, each with its specific movement allowed. An admissible trajectory must start and end at the λφ axis, which can only be the case for a trajectory such as the lower solid curve. In this case, Hφ ≥ 0.
5.2. Parameter shifts and changes in the utility-maximising time
paths
Here we will examine the effects of a parametrical shift on the
utility-maximising time paths of behaviour and health. Consider two
individuals who face different sensitivities to deviations from B*, φ
and φ + dφ, but who are otherwise identical. The difference in their
behaviour, and resulting health, can be examined as the change in
behaviour that one of the individuals would implement if faced with
the sensitivity of the other individual. In what follows, this is what
we have in mind when we examine the effects of a parametrical
shift.
A shift in a parameter will perturb the entire utility-maximising
time path of behaviour and health capital.27 However, the phase portrait introduced in the previous section is not invariant to parametric
changes, which means that different trajectories corresponding to
different parametrical values cannot readily be compared within the
same diagram. Therefore, we will apply the technique developed
by Oniki (1973), which employs the equations of motion of the health
stock and its shadow price to investigate the effect of a change in
an exogenous parameter for the time paths.28,29 The perturbation
of the time paths may be discerned from phase diagrams similar
to the one utilised above, but with the axes measuring variation in
the shadow price of health capital and in the health stock, in relation to the unperturbed situation.
27 The shift can occur at t ∈ [0, T). A shift at t = 0 means that the individual faces a
different optimisation problem than before the shift; a shift at t > 0 means that the
individual re-optimises his or her behaviour at time t, taking the health stock, H(t),
as given.
28
This method was developed by Oniki (1973) and has previously been used for
analysing models similar to ours (see, for instance, Ehrlich and Chuma, 1990; and
Eisenring, 1999).
29
Oniki’s method applies the variational differential equation approach to comparative dynamics analysis. Essentially, and in terms of our model, this method consists
of differentiating the equations for H (t ) and λ (t ), with respect to the parameter,
x, of interest. The equation system obtained can then be used for inferring the signs
of Hx(t) and λx(t), i.e., the change in H and λ at time t that results from the change
in x. The sign of Bx is deduced from the first-order condition (8) and/or the health
investment production function. A more detailed account of this approach is provided in the Appendix; a further extension and detailed calculations are presented
in the online supplement.
We will examine the effect on the utility-maximising time paths
of a shift in one of the parameters, φ (the rate at which deviations
from the physiologically optimal level of B are punished). Comparative dynamics results for the other parameters are presented in
the online supplement. The analysis will be performed in two steps,
starting with the examination of the change in the entire timepaths of the shadow price of health capital and of the health stock,
and followed by the analysis of the changes in the behaviour, B.
Prediction 5. Both low- and high-exerting individuals are healthier
the higher is the sensitivity to deviations from the level of healthbehaviour activity at which there is no punishment in terms of reduced
health.
The phase diagram in Fig. 4 illustrates admissible time paths of
the (perturbed) shadow price of health capital and the stock of health
capital induced by a shift in φ, for a low-exerting individual, i.e., an
under-exerting type individual, behaving at a level below B*, and
for a high-exerting individual, i.e., an over-exerting type individual. The corresponding phase diagram for the moderately exerting
individual, i.e., an under-exerting type individual, behaving at a level
above B*, will be examined in a following section.
The diagram is constructed in the perturbed, [ Hϕ (t ) , λϕ (t )],
space.30 It is divided into four sections by the H ϕ = 0 and the λϕ = 0
loci, which have the same significance as in an ordinary phase
diagram. Each section allows for a specific motion of the perturbed health stock and the shadow price, Hϕ (t ) , λϕ (t ) . It is clear that
an admissible trajectory must start and end at the vertical axis, since
both the initial and end-point health stocks are fixed ( Hϕ (0) = 0 and
Hϕ (T ) = 0 ). This can only be achieved if the trajectory starts above
the H ϕ = 0 locus and ends below it, with all motions taking place
in the fourth quadrant. Such a trajectory implies that the perturbed shadow price of health capital is lower than the unperturbed
shadow price, λφ < 0, for all t, and that the perturbed health stock
is larger than the unperturbed stock, Hφ > 0, over the entire life cycle
(0, T ) . The intuition is straightforward. A higher φ means that a
smaller increase in the B-behaviour is required in order to produce
30
The phase diagram could – in principle – be constructed in the [ Hϕ , Bϕ ] space.
That alternative, however, is less intuitive for understanding behavioural changes.
18
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
one additional unit of health capital, i.e., the marginal cost of health
capital will decrease. This will lead to an increase in the stock of
health capital and a decrease in its shadow price.31,32
Further, the perturbation of the shadow price of health capital
will increase over the entire life cycle, i.e., λϕ < 0 , which will gradually diminish health investments. At some point in time, the
perturbation of the health stock will start to decrease, in order to
reach Hmin at t = T. Thus, an upward shift in φ will make both the
under-exerting type, who behaves at low levels of B, and the overexerting type, healthier. So, how will each type alter his or her utilitymaximising time-path of the B-behaviour? Let us examine this,
beginning by the under-exerting type.
Prediction 6. An increase in the sensitivity to deviations from the
health-behaviour level at which there is no punishment in terms of
reduced health makes the low-exerting type exert more of the behaviour
in an initial phase and less thereafter but makes the high-exerting type
exert less in the initial phase and more thereafter.
We start with examining the low-exerting individual. Clearly, the
instantaneous effect has to be an increase in gross health investments, since the perturbed health stock is larger than the
unperturbed stock for all points in time, except at the starting and
end points. It is straightforward to verify that the level of the
B-behaviour will continue to be higher, compared to the initial situation, until some point in time, t < T, at which the perturbed
B-behaviour equals the unperturbed behaviour.33 After that, B necessarily falls below the unperturbed level, since otherwise the “death”
health stock would not be reached at t = T. In order to understand
how the actual behavioural choices that the individual makes change,
we need to make use of the fact that the instantaneous marginal
utility of B must equal its marginal health costs (or benefits) at all
points in time (Equation 8′). Thus, given the perturbed time-paths
of the shadow price of health capital and the health stock, the corresponding behavioural changes can be understood using the same
set-up as in Fig. 2.
The behavioural changes may be illustrated graphically as the
combination of a direct effect – the upward shift in φ – and an indirect effect – the change in the shadow price of health capital, λ
(see Fig. A3). The direct effect of the upward shift in φ is to decrease the physiologically optimal level of B, making any given
behavioural deviation from B* more costly (the marginal cost curve
becomes more steep). This is shown as the leftward rotation of the
solid marginal cost curve around the point λ (0) , B* to the dotted
curve. It is clear that this strengthens the incentives for exerting the
B-behaviour at a level closer to B*. The indirect effect on behaviour
is illustrated by the rightward rotation of the dotted marginal cost
curve around the new point of physiologically optimal behaviour,
1
(the dotted line in Fig. A3) to the dashed marginal cost
2ϕ̂
curve. The instantaneous change in B is reflected by the dashed curve.
Over time, the marginal cost curve will shift further rightwards and,
at some point in time, indicate a level of B below the unperturbed
level.
B* +
31 The marginal health benefit of B is 1 − 2ϕ ( B − B* )
, which increases when φ increases when B* > B.
1
32 1 − 2ϕ ( B − B* )
+ B* < B .
decreases when φ increases when
2⋅ ϕ
33
As long as the difference between the perturbed and unperturbed health
stocks increases, gross health investments must be higher after the shift.
Differentiating the health-production function, see Equation (4), yields:
2
dI = (1 − 2ϕ (B − B*)) dB − (B − B*) dϕ . For the under-exerting type behaving at B* > B
we have, setting dI > 0: dB >
(B − B*)2
dϕ > 0 , which must be true as long as the
(1 − 2ϕ (B − B*))
perturbations of the health stock increase.
Proceeding as in the previous case for the over-exerting type, it
is easy to verify that the level of the B-behaviour decreases instantaneously and for an initial time period, compared to the initial
situation, after which the B-behaviour necessarily is raised above
the unperturbed level, in order to reach the “death” health stock
at t = T.34 The change in the behavioural choices is illustrated in
Fig. A4.
Prediction 7. Moderately exerting individuals are less healthy the higher
is the sensitivity to deviations from the health-behaviour level at which
there is no punishment in terms of reduced health.
It remains to examine the impact of the shift in φ for health and
health behaviour of the under-exerting type behaving at moderate levels of B (B > B*). The [ Hϕ , λϕ ]-phase diagram for this case is
illustrated in the online supplement, Figure 4. Again, an admissible trajectory must start and end at the vertical axis, since initial
and end-point health levels are fixed ( Hϕ (0) = 0 and Hϕ (T ) = 0 ). In
this case, this can only be achieved if the trajectory starts below the
H ϕ = 0 locus and ends above it, with all motions taking place in
the second quadrant. Such a trajectory implies that the perturbed
shadow price of health capital is higher than the unperturbed
shadow price, λφ > 0, for all t, and that the perturbed health stock
is smaller than the unperturbed stock, Hφ < 0, over the entire life
cycle (0, T ) . The intuition is straightforward. A higher φ here means
that a larger increase in the B-behaviour is required in order to
produce one additional unit of health capital, i.e., the marginal cost
of health capital will increase. This will lead to a decrease in the
stock of health capital and an increase in its shadow price.35
Further, the perturbation of the shadow price of health capital
increases over the entire life cycle, λϕ > 0 , which gradually strengthens the incentives for making health investments. As in previous
cases, at some point in time, the perturbation of the health stock
will start to decrease, in order to reach Hmin at t = T. Thus, an upward
shift in φ will make the under-exerting type, who behaves at moderate levels of B, less healthy. In this case, the changes in the level
of the B-behaviour do not mirror the perturbations of the health
stock, since the upward shift in φ decreases both the demand for
health capital and the supply of health investments.36
The changes in optimal time paths that result from changes in
the exogenous parameters are summarised in Table 1.
6. Discussion
In this paper, we extended the health-capital model to include
health-related behaviours that influence health non-monotonically.
In comparison with the original Grossman model, the distinguishing features of our model are (1) the introduction of (a) a threshold
level of behaviour, above or below which the individual experiences penalties in terms of lower health, and (b) an individual- or
behavioural-specific sensitivity of deviating from the threshold level,
and (2) the assumption that health-related behaviours not only affect
the individual’s production of health but also provide instantaneous utility. Our model belongs to a family of models characterised
by a behaviour yielding instantaneous utility while adding to a stock
that influences future optimal choices, i.e., the same family of models
as the models developed by Becker and Murphy (1988) and Dockner
and Feichtinger (1993), respectively.
34
35
In this case we have: dI > 0, which gives dB <
(B − B*)2
dϕ < 0 .
(1 − 2ϕ (B − B*))
1 − 2ϕ (B − B*) decreases when φ increases when B* < B < 1 + B* .
2⋅ϕ
See footnote 26. In this case we have the condition that dI < 0, which gives
.
(B − B*)2
dB <
dϕ > 0
(1 − 2ϕ (B − B*))
36
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
19
Table 1
Summary of the comparative dynamics results.* The table reports the changes in the optimal time paths that result from an increase in parameter x at t = 0. Most effects are
unambiguous, taken over a part of the time interval [0, T ]. This is so since both starting and end points are fixed.
H0
δ
q
w
φ
ρ
B<B
1
B* < B < B* +
2ϕ
1
B* +
<B
2ϕ
*
H x (t )
λx(t)
Bx(t)
Over (under)
exerting
Over (under)
exerting
Under
exerting
Over
exerting
Under
exerting
Over
exerting
Under
exerting
Under
exerting
Over
exerting
Over (under)
exerting
H H0 > 0 a
λ H0 < 0
BH0 > (< ) 0
Hδ < 0 b
λδ > 0
Bδ > (< ) 0 c
H q < 0b
λq > 0
Bq < 0 c
Hq > 0b
λq < 0
B q < 0c
Hw ≶ 0
λw ≷ 0
Bw ≷ 0
Hw > 0 b
λw < 0 c
Bw < 0 c
H φ > 0b
λφ < 0
B φ > 0c
Hφ < 0b
λφ > 0
Bφ < > 0
H φ > 0b
λφ < 0
Bφ < 0c
H ρ < 0b
λρ > 0c
Bρ > (< )0c
* All calculations are shown in the online supplement.
t ∈[0, T ) .
b t ∈ (0, T )
.
c t ∈ [0, t1 ] t1 < T . Notice, t1 may differ between individual types and parameters.
a
Box 2Summary of findings.
Prediction 1: Behaviours that influence health non-monotonically are exerted at levels that are either lower or higher than what would
be physiologically optimal.
Prediction 2: The stronger the preferences for the health-behaviour activity, the lower its unit cost and the higher the sensitivity to
deviations from the level at which there is no punishment in terms of reduced health, the more likely the individual would over-exert
the activity, i.e., utilise it more than what is physiologically optimal.
Prediction 3: For given individual characteristics and prices, the individual will not change his or her type from over-exerting to underexerting or vice versa during his or her life time.
Prediction 4: Individuals endowed with relatively high health levels may initially behave more healthy but eventually behave less healthy
over time, while individuals endowed with low health levels always behave less healthy over time.
Prediction 5: Both low- and high-exerting individuals are healthier the higher is the sensitivity to deviations from the level of healthbehaviour activity at which there is no punishment in terms of reduced health.
Prediction 6: An increase in the sensitivity to deviations from the health-behaviour level at which there is no punishment in terms of
reduced health makes the low-exerting type exert more of the behaviour in an initial phase and less thereafter but makes the highexerting type exert less in the initial phase and more thereafter.
Prediction 7: Moderately exerting individuals are less healthy the higher is the sensitivity to deviations from the health-behaviour level
at which there is no punishment in terms of reduced health.
Main model predictions are summarised in Box 2. Two major
types of individuals were identified, those who exert less and those
who exert more than the physiologically optimal health behaviour
for that person. The first type of individual is similar to the individual in the original Grossman model, except that in the original
model there would be no satiation point for investments in health.
Our model allows for a variety of analyses of the relationship
between individual characteristics, and market factors, and choices
of behaviour over the life cycle. In the preceding sections we focused
on the significance of the sensitivity of deviating from the threshold level. We demonstrated that the higher the sensitivity, the more
likely the individual would be to exert the behaviour above the physiologically optimal level, ceteris paribus. The reason is that as the
physiologically optimal level decreases the marginal utility of the
behaviour becomes more likely to equal its marginal cost at a level
which is larger than the physiologically optimal level. Thus, individuals who face a high sensitivity are more likely to be overexerting individuals, and, similarly, behaviours that are characterised
by a high sensitivity are more likely to be exerted at levels above
the physiologically optimal level.
We demonstrated that the higher the sensitivity of deviating from
the threshold level, the higher would health be over the life cycle,
while behaviour would be closer to the physiologically optimal level
for individuals who exert either low or high (over-exerting) levels
of behaviour. The explanation for this is that the marginal productivity for producing investments in health is increasing in the
sensitivity in these two cases.
We believe that the opportunities to empirically estimate associations between these variables and/or to test the predictions are
good – at least in the Nordic countries, where various relevant administrative registers can be linked together fairly easily and also
be linked to population questionnaire data.
The dynamic nature of the model requires several empirical observations over time for the same individual (Grossman, 2000); this
is available in Sweden for a fairly large population sample, for health,
for health-related behaviours, and for background variables.
Relevant health-related behaviours are those that have a U-shaped
form in terms of impact on health, i.e. most behaviours except
smoking (which has a definitely negative impact irrespective of dose).
Details on alcohol consumption, physical-exercise habits, BMI, subjective health, and most individual background variables (age, gender,
education, profession, income, wealth, civil status, family, labourmarket attachment, etc.) can be collected from a biennial population
survey (Survey of Living Conditions in Sweden). Background variables are also available in registers that can be linked to the individual
panel data of the survey, using the unique individual personal identification number. Registers also contain data on the utilisation of
welfare services and mortality. Furthermore, it is possible to trace
the personal identification numbers of parents (and, hence, also
grandparents) and add relevant register information on them if required. There is no indicator of individual time preferences based
on hypothetical questions available in the Swedish Survey of Living
Conditions, so proxies would have to be used: for instance, growth
of consumption (Carroll and Summers, 1991; Lawrence, 1991);
savings rate (Komlos et al., 2004); or assets/liabilities (Borghaus and
Golsteyn, 2006).
Most of the above would also be required in order to estimate
the dynamic version of the original Grossman model (Bolin et al.,
20
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
2002a). In addition, there is the need to find empirical counterparts (a) to what is physiologically optimal for a specific healthrelated behaviour, (b) to the individual’s degree of sensitivity to
deviations from the threshold level of behaviour, and (c) to the sensitivity to deviations embodied in the specific health behaviour. It
should be emphasised that all these three items are certainly individually determined. They cannot be observed, however, so proxies
have to be used instead. For the physiologically optimal levels, averages from the literature would probably have to be used. The same
goes for health-behaviour-specific sensitivity to deviations. For the
individual sensitivity to deviations in general, previous history on
the individual’s use of health services and/or the history of parents’
use of health services and/or parents’ (grandparents’) length of life,
cause of death, etc. might serve as indicators.
The model has a number of qualities. It represents a true novel
theoretical approach to individual health behaviour. It is also an important approach; it incorporates the fact, observed in a number
of epidemiological studies, that several health behaviours have an
inverted U-shaped impact on health. It provides a framework for
understanding behaviours that are sub-optimal in relation to health,
i.e. behaviours that are either below or above a physiologically
optimal level. The model produces a number of theoretical predictions, which in principle are empirically testable. There also seem
to be ample opportunities for empirical estimations and/or tests in
practice. In some countries, for instance, there already exist individual data registers and/or surveys that can be used for this purpose.
Policy-relevant conclusions would depend very much on to what
extent theoretical predictions will be confirmed and associations
estimated in empirical analyses.
Thus, our model extends the theoretical demand-for-health
framework, and it does so providing a number of new empirically
testable predictions. Nevertheless, even though the model may
seem complex, there are still some assumptions (or abstractions
from “the real world”) that might be worthwhile to try to relax in
future theoretical work. One rather obvious extension would be
to let the production of health have two (types of) inputs, one
which also provides instantaneous utility as here (alcohol, physical exercise, etc.) and one which does not (“medical care”). Another
would be to introduce asymmetries in the sensitivity to deviations from the threshold level of health behaviour; it might be
more detrimental to health to exert more rather than less health
behaviour than physiologically optimal – or vice versa. A third
would be to assume that health behaviours differ regarding the
extent to which produced gross health investments are transformed into net health investments37; Bolin et al. (2014) develop a
theoretical model with the main objective of analysing the importance of such discrepancies. Additional potential extensions are:
(a) to introduce risk and/or uncertainty, for instance, by assuming
that individuals have less than complete accurate knowledge
about his or her sensitivity to deviations from the threshold level
of behaviour (which might be related to educational level), (b) to
allow for exogenous discrete “shocks”, which might make an
over-exerting type individual switch to an under-exerting type or
vice versa – something which is impossible in our present model,
or (c) by changing end conditions. It might, finally, also be worthwhile to examine what changes have to be introduced in the
model in order to accommodate endogenous behaviour that makes
an under-exerting type individual turn into an over-exerting one
and vice versa. Potential extensions are plenty.
Acknowledgements
We are grateful for extensive and helpful comments from three
anonymous referees. We thank Lance Brannman, Martin Forster,
Michael Grossman, and Robert Kaestner for insightful comments
and useful suggestions.
Appendix
In the Appendix we show the calculations regarding: (1) the stability properties of steady state, (2) the shape of the stationary loci, (3)
the principles for steady-state comparative statics calculations, and (4) the comparative dynamics analysis. For brevity, time-arguments
have, wherever appropriate, been omitted.
Stability
In this section we will perform the necessary calculations for the general utility function, and provide the simplified results which follow
when the utility function is additively separable and quasilinear (ASQU), as specified in Section 5. A sufficient condition for the steady
state to be locally well defined is that the Jacobian determinant of the equation system comprised of Equation (4), and the equation obtained by taking the time derivate of Equation (8), evaluated in steady state, is not zero (the implicit function theorem). We have:
2
⎡
π
π
w
π
πw ⎤
⎡
⎛π⎞ ⎤
λ (1 − 2ϕ (B − B*)) − B ⎢λ 2ϕ − U BB + U BZ + U ZB − U ZZ ⎜ ⎟ ⎥ + H ⎢U BH − U BZ τ H − U ZH − U ZZ
τ H = 0 . Solving for B gives the second equation:
⎝ p⎠ ⎦
p
p
p
p
p p ⎥
⎣
⎣
⎦
D
F≶0
E >0
λ D + HF
.
B =
E
37
This possibility was suggested by one reviewer.
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
21
Jacobian matrix, pertaining to the equations for
H and
B and evaluated in steady state is:
−δ
D
D
−δ
⎡
⎤ ⎡
⎤ 38
def
. For the additively separable and quasilinear utility function that
H λ B D + λ DB + H B F + HF
B ⎥ = ⎢ λ H D + H H F λ B D + H B F ⎥
J de = ⎢ λ H D + H H F + HF
⎢
⎥ ⎢
⎥
⎢⎣
⎥⎦ ⎢⎣
⎥⎦
E
E
E
E
The
−δ
⎡
def
we employ we have: J de = ⎢ λ H D
⎢
⎢⎣ λ 2ϕ − vBB
D
⎤
⎥
λ B D ⎥ , since E = λ 2ϕ − vBB and F = 0.
λ 2ϕ − vBB ⎥⎦
w
Expressions for λ H and λ B are obtained by differentiating Equation (7): λ = −U H + U z τ H + λ (δ + ρ ) – w.r.t. H and B, respectively, using
p
π
−U B + U z
p
λ=
:
(1 − 2 ⋅ ϕ (B − B*))
2
2
w
w
w
w
⎛w ⎞
⎛w ⎞
λ H = −U HH − U HZ τ H + U ZH τ H − U ZH ⎜ τ H ⎟ + U Z τ HH = −U HH − U ZH ⎜ τ H ⎟ + U Z τ HH >< 0 ; for the ASQU utility function we have:
⎝p ⎠
⎝p ⎠
p
p
p
p
w
λ H = −U HH + k τ HH > 0 .
p
2
π
π
⎛π⎞
−U BB + 2U BZ − U ZZ ⎜ ⎟
UB − Uz ⋅
⎝ p⎠
π
w
πw
p
p
λ B = −U HB − U HZ + U ZB τ H − U ZZ
τ H + (δ + ρ ) ⋅
− 2ϕ (δ + ρ )
>< 0 ; for the ASQU utility function
p
p
p p
(1 − 2 ⋅ ϕ (B − B*))
(1 − 2 ⋅ ϕ (B − B *))2
π
UB − k
p
, implying that λ B D = 2ϕ (δ + ρ ) λ > 0.
we have: λ B = −2ϕ (δ + ρ )
(1 − 2 ⋅ ϕ (B − B*))2
def
The determinant of the Jacobian matrix evaluated in steady state is J de =
sufficient condition for saddle-point stability. When B =
−δ
λH D
λ 2ϕ − vBB
D
1
=−
(δλ B D + D2λ H ) < 0, which is a
λB D
λ 2ϕ − vBB
λ 2ϕ − vBB
1
+ B* – an optimally exerting type individual – the determinant of the Jaco2⋅ϕ
bian matrix is zero, and the trace is strictly negative, i.e., one eigenvalue is zero and the other is strictly negative. This implies a sink.
Comparative statics of steady state
Introducing the notation: h ( H , B ) ≡ B − ϕ ( B − B*)2 − δ ⋅ H , and g ( H , B ) ≡ λ D the differentiated system becomes the effects of a change in
an exogenous parameter, x, on steady-state levels of health capital and the B-behaviour, He and Be, can be obtained by differentiating the
equation system ⎣⎡ H ( H e, Be ) = 0, B ( H e, Be ) = 0 ⎦⎤. The differentiated system becomes:
⎛ dH e ⎞
⎛ hH hB ⎞ ⎜ dx ⎟ ⎛ −hx ⎞
.
⎟=
⎜
⎜⎝ g
g B ⎟⎠ ⎜ dBe ⎟ ⎜⎝ − g x ⎟⎠
H
⎝ dx ⎠
Js
In steady state, we have: g B = λ B D e , g H = λ H D e , and g x = λ x D e . Applying Cramer’s rule the solution is obtained as:
B= B
B= B
B= B
D ⎞
⎛ −hx
⎜ − g λ D ⎟
x
B
dH
⎛
⎞ ⎜
⎟
J sd
⎜ dx ⎟ ⎜
⎟
=
⎜
⎟ ⎜
−hx ⎟⎟
⎜ dB ⎟ ⎜ hH
.
⎝ dx ⎠ ⎜ λ D −g ⎟
⎜ H d x⎟
⎠
⎝
Js
Complete calculations are shown in the online supplement.
38
Note, in steady state H = B = 0 , which implies that λ = 0 .
22
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Phase diagram
Shape of the H = 0 locus
An explicit expression can be obtained for the H = 0 locus. Rearranging Equation (4) yields: B =
parabola having a vertex ( H max =
1 + 2B*
1 + 4 B *ϕ δ
ϕ
±
− H , which is a
2
4ϕ 2
ϕ
1
1 + 4 B *ϕ
+ B* , and intersections with the B-axis at B = 1 + B* ± 1 + 4 B*ϕ .
) at B =
2ϕ
4δϕ
2ϕ
4ϕ 2
Slope of the B = 0 locus
λ D
The slope of the B = 0 loci is found by implicit differentiation of B =
= 0 (see above). For the under- and over-exerting type indiE
dB
λ D
= − H > (< ) 0 when (1 − 2ϕ ( B − B*)) < ( > ) 0. Thus, the
viduals D ≡ (1 − 2ϕ ( B − B*)) ≠ 0 , in which case the implicit function theorem yields:
dH
λB D
loci have the shape of a hyperbola with one branch above – an over-exerting type individual – and one below – an under-exerting type
individual – the horizontal line at B (t ) =
1
+ B* .39 The upper branch is increasing, while the lower is decreasing. The curvature is analysed
2ϕ
in the online supplement.
Direction of trajectories
The direction of a trajectory is obtained in the following way. Using the notation introduced above of h(H, B) and g(H,B), we have, first,
for the motion of H: starting at the H = 0 locus and moving slightly to the right (left), i.e., increasing (decreasing) H, will place us at a
H < ( > ) 0 trajectory, since hH = −δ. Similarly, for movements in the vertical direction we have: g H = λ H (1 − 2 ⋅ ϕ ⋅ ( B (t ) − B*)) > (< ) 0 when
1
1
B < (>)
+ B* . Thus, B > (< ) 0 to the right of the B = 0 locus when B < ( > )
+ B* .
2⋅ϕ
2⋅ϕ
Comparative dynamics
We apply special case (b) of Oniki’s theorem by Oniki (1973, pp. 276–282) when computing the impact of a change in an exogenous
parameter, x, on the optimal time paths of H and B. Following the notation used by Eisenring (1999) we obtain for a change in the exogenous parameter, x:
⎛ ∂ H
∂ ⎛ H x (t; x )⎞ ⎜ ∂H
=⎜
∂t ⎝⎜ λ x (t; x ) ⎠⎟ ⎜ ∂ ⎝ ∂H λ
∂ ⎞
⎛ ∂ H ⎞
H
∂λ ⎟ ⎛ H x (t; x )⎞ ⎜ ∂x ⎟
+⎜
⎟⋅
⎟ . The signs of the elements of the Jacobian matrix, the first term on the right-hand side,
∂ ⎟ ⎝⎜ λ x (t; x ) ⎠⎟ ⎜ ∂ ⎟
λ⎠
λ
⎝ ∂x ⎠
∂λ
⎛ − +⎞
. The phase diagrams are constructed using this information and the location of the H x = 0 and λ x = 0 loci. For
can be shown to be ⎜
⎝ + + ⎟⎠
H (t; ϕ )⎞ ⎛ − + ⎞ ⎛ Hϕ (t; ϕ )⎞ ⎛ −⎞
. Setting the right-hand side equal to
a change in φ, we have: ∂ ⎛ ϕ
=
⋅
+
∂t ⎝⎜ λϕ (t; ϕ ) ⎠⎟ ⎜⎝ + + ⎟⎠ ⎜⎝ λϕ (t; ϕ ) ⎠⎟ ⎜⎝ 0⎟⎠
⎛ 0⎞ gives the isoclines. Detailed calcu⎜⎝ 0⎟⎠
lations are shown in the on-line supplement. Notice that the elements of the Jacobian matrix shift over time and, hence, the isoclines
will also shift over time. However, the qualitative properties of the phase diagram will not change.
1
39
+ B* , which is a third branch of
For the under- and over-exerting types, we have 1 − 2 ⋅ ϕ ⋅ (B (t ) − B*) ≠ 0 ; and for an optimally exerting type individual, we have B (t ) =
2ϕ
the B (t ) = 0 locus.
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
23
Fig. A1. Illustration of the relationship between behaviour, B, and the amount of health investment, I. The parabola is the graph of the function I (t ) = B (t ) − ϕ ⋅ (B (t ) − B*) .
The maximum influence on health attainable through the behaviour, B, is at the inflexion point of the parabola. The solid-line parabola corresponds to the value φ, while
the high-location dotted parabola corresponds to the value φ’ < φ, and the low-location dotted parabola corresponds to the value φ’’ > φ.
2
E
F
Over exerting type
Under exerting type
C
B
D
A
Fig. A2. Illustration of admissible trajectories in state-control space for high and low initial health. All plotted trajectories fulfil conditions (4) and (7). All trajectories fulfil
the end-point condition, H (T ) = H T , and, hence, are possible solutions to the optimisation problem.
24
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Fig. A3. Illustration of the behavioural effects of an upward shift in φ, for the under-exerting individual behaving at B < B*. The first shift of the marginal cost curve, from
the solid to the dotted curve, is due to the direct effect of a change in φ. This is the leftward rotation around the point (−λ (0) , B*) . The second shift, from the dotted to the
dashed curve, is due to the indirect effect through the instantaneous change in λ(0) to λ̂ (0). This is illustrated by the rightward rotation around the new physiologically
1
. The total instantaneous change in behaviour is the distance between B and B̂ . For successively smaller shadow prices the dashed line will rotate
optimal point, B* +
2ϕ̂
further rightward, implying lower levels of B. The dashed-dotted curve illustrates the behaviour at t = T.
Fig. A4. Over-exerting individual. Illustration of the total shift of the marginal cost curve induced by an increase in φ. The first shift of the line – from the solid to the dotted
line – is due to the direct effect of a change in φ. This is the leftward rotation around the point (−λ, B*) . The second shift of the line – from the dotted to the dashed line –
1
. The total change in
is due to the indirect effect through the change in λ. This is illustrated by the leftward rotation around the new physiologically optimal point, B* +
2ϕ̂
behaviour is B to B̂ .
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Appendix: Supplementary material
Supplementary data to this article can be found online at
doi:10.1016/j.jhealeco.2016.08.003.
References
Arrow, K.J., 1963. Uncertainty and the welfare economics of medical care. The
American Economic Review 53, 941–973.
Asano, T., Shibata, A., 2011. Risk and uncertainty in health investment. European
Journal of Health Economics 12, 79–85.
Becker, G.S., Murphy, K.S., 1988. A theory of rational addiction. The Journal of Political
Economy 96, 675–700.
Bize, R., Johnson, J.A., Plotnikoff, R.C., 2007. Physical activity level and health-related
quality of life in the general adult population: a systematic review. Preventive
Medicine 45, 401–415.
Bolin, K., Jacobson, L., Lindgren, B., 2001. The family as the producer of health – when
spouses are Nash bargainers. Journal of Health Economics 20, 349–362.
Bolin, K., Jacobson, L., Lindgren, B., 2002a. The demand for health and health
investments in Sweden 1980/81, 1988/89, and 1996/97. In: Lindgren, B. (Ed.),
Individual Decisions for Health. Routledge, London, pp. 93–112.
Bolin, K., Jacobson, L., Lindgren, B., 2002b. The family as the producer of health –
when spouses act strategically. Journal of Health Economics 21, 475–495.
Bolin, K., Jacobson, L., Lindgren, B., 2002c. Employer investments in employee health
– Implications for the family as health producer. Journal of Health Economics
21, 563–583.
Bolin, K., Lindgren, B., Lindström, M., Nystedt, P., 2003. Investments in social capital
– implications of social interactions for the production of health. Social Science
& Medicine 56, 2379–2390.
Bolin, K., Lindgren, B., Liljas, B., 2014. Individual technologies for health – the
implications of distinguishing between the ability to produce health investments
and the capacity to benefit from those investments. Working Paper No 587,
Department of Economics, University of Gothenburg.
Borghaus, L., Golsteyn, B., 2006. Time discounting and the body mass index: evidence
from the Netherlands. Economics and Human Biology 4, 29–61.
Buxton, O.M., Marcelli, E., 2010. Short and long sleep are positively associated with
obesity, diabetes, hypertension, and cardiovascular disease among adults in the
United States. Social Science & Medicine 71, 1027–1036.
Caputo, M.R., 2005. Foundations of Dynamic Economic Analysis – Optimal Control
Theory and Applications. Cambridge University Press, Cambridge.
Carbone, J., Kverndokk, S., Individual investments in education and health, HERO
Working Paper 2014:1, University of Oslo.
Carroll, C.D., Summers, L.H., 1991. Consumption growth parallels income growth.
In: Bernheim, B.D., Shoven, J.B. (Eds.), National Savings and Economic
Performance. University of Chicago Press, Chicago, pp. 305–343.
Chang, F., 1996. Uncertainty and investment in health. Journal of Health Economics
15, 369–376.
Chang, Y.M., 2007. Transfers and bequests: a portfolio analysis in a Nash game. Annals
of Finance 3, 277–295.
Chang, Y.M., 2009. Strategic altruistic transfers and rent seeking within the family.
Journal of Population Economics 22, 1081–1098.
Chang, Y.M., Weisman, D.L., 2005. Sibling rivalry and strategic parental transfers.
Southern Economic Journal 71 (4), 821–836.
Chiang, A.C., 1992. Elements of Dynamic Optimization. McGraw Hill, Singapore.
Colditz, G.A., 1999. Economic costs of obesity and inactivity. Medicine and Science
in Sports and Exercise 31, S663–S667.
Colditz, G.A., 2000. Illnesses caused by smoking cigarettes. Cancer Causes and Control
11, 93–97.
Cropper, M.L., 1977. Health, investment in health, and occupational choice. The Journal
of Political Economy 85, 1273–1294.
Cutler, D.M., Gleaser, E., 2005. What explains differences in smoking, drinking, and
other health-related behaviours. The American Economic Review 95 (2), 238–242.
Dal Grande, E., Gill, T., Wyatt, L., Chittleborough, C.R., Phillips, P.J., Taylor, A.W., 2009.
Population attributable risk (PAR) of overweight and obesity on chronic diseases:
South Australian representative, cross-sectional data, 2004–2006. Obesity
Research & Clinical Practice 3, 159–168.
Dardanoni, V., Wagstaff, A., 1987. Uncertainty, inequalities in health and the demand
for health. Journal of Health Economics 6, 283–290.
Dardanoni, V., Wagstaff, A., 1990. Uncertainty and the demand for medical care.
Journal of Health Economics 9, 23–38.
Dockner, J.E., Feichtinger, G., 1993. Cyclical consumption patterns and rational
addiction. The American Economic Review 83 (1), 256–263.
Doll, R., Peto, R., 1976. Mortality in relation to smoking: 20 years’ observations on
male British doctors. British Medical Journal 2, 1525–1536.
Doll, R., Peto, R., Wheatley, K., Gray, R., Sutherland, I., 1994. Mortality in relation to
smoking: 40 years’ observations on male British doctors. British Medical Journal
309, 901–911.
Dowie, J., 1975. The portfolio approach to health behaviour. Social Science and
Medicine 9, 619–631.
Ehrlich, I., 2000. Uncertain lifetime, life protection, and the value of life saving. Journal
of Health Economics 19, 341–367.
Ehrlich, I., 2001. Erratum to “ Uncertain lifetime, life protection, and the value of life
saving. Journal of Health Economics 20, 459–460.
25
Ehrlich, I., Chuma, H., 1990. A model of the demand for longevity and the value of
life extensions. The Journal of Political Economy 98, 761–782.
Eisenring, C., 1999. Comparative dynamics in a health investment model. Journal
of Health Economics 18, 655–660.
Forster, M., 2001. The meaning of death: some simulations of a model of healthy
and unhealthy consumption. Journal of Health Economics 20, 613–638.
Galama, T., 2011. A contribution to health capital theory. Working Paper WR-831.
Rand Corporation, Santa Monica, CA.
Galama, T., Kapteyn, A., 2011. Grossman’s missing health threshold. Journal of Health
Economics 30, 1044–1056.
Galama, T., van Kippersluis, H., 2013. Health inequalities through the lens of
health-capital theory: issues, solutions, and future directions. Research on
Economic Inequality 21, 263–284.
Galama, T., Kapteyn, A., Fonseca, R., Michaud, P.C., 2013. A health production model
with endogenous retirement. Health Economics 22, 883–902.
Garrett, N.A., Brasure, M., Schmitz, K.H., Schultz, M.M., Huber, M.M., 2004. Physical
inactivity. Direct cost to a health plan. American Journal of Preventive Medicine
27, 304–309.
Grossman, M., 1972a. The Demand for Health: A Theoretical and Empirical
Investigation. Columbia University Press for the National Bureau of Economic
Research, New York.
Grossman, M., 1972b. On the concept of health capital and the demand for health.
The Journal of Political Economy 80, 223–255.
Grossman, M., 2000. The human capital model of the demand for health. In: Culyer,
A., Newhouse, J. (Eds.), Handbook of Health Economics. Elsevier, Amsterdam, pp.
347–408.
Grönbaeck, M., 2009. The positive and negative health effects of alcohol – and the
public health implications. Journal of Internal Medicine 265, 407–420.
Grönbaeck, M., Deis, A., Becker, U., Hein, H.O., Schnohr, P., Jensen, G., Borch-Johnsen,
K., Sörensen, T.I.A., 1998. Alcohol and mortality: is there a U-shaped relation in
elderly people? Age and Ageing 27, 739–744.
Howatson, G., van Someren, K.A., 2008. The prevention and treatment of exerciseinduced muscle damage. Sports Medicine 38, 483–503.
Hurley, L.L., Taylor, R.E., Tizabi, Y., 2012. Positive and negative effects of alcohol and
nicotine and their interactions: a mechanistic review. Neurotoxicity Research
21 (1), 57–69.
Ippolito, P.M., 1981. Information and the life cycle consumption of hazardous goods.
Economic Inquiry 19 (4), 529–558.
Jacobson, L., 2000. The family as producer of health – An extension of the Grossman
model. Journal of Health Economics 19, 611–637.
Ji, L.L., 2001. Exercise at old age: does it increase or alleviate oxidative stress? Annals
of the New York Academy of Sciences 928, 236–247.
Knuth, A.G., Hallai, P.C., 2009. Temporal trends in physical activity: a systematic
review. Journal of Physical Activity and Health 6, 548–559.
Komlos, J., Smith, P., Bogin, B., 2004. Obesity and the rate of time preference: is there
a connection? Journal of Biosocial Science 36, 209–219.
Konrad, K.A., Lommerud, K.E., 2000. The bargaining family revisited. Canadian Journal
of Economics 33, 471–487.
Konrad, K.A., Künemund, H., Lommerud, K.E., Robledo, J., 2002. Geography of the
family. The American Economic Review 92, 981–998.
Lawrence, E.C., 1991. Poverty and the rate of time preference: evidence from panel
data. The Journal of Political Economy 99, 54–77.
Lee, D., Lavie, C.L., Vedanthan, R., 2015. Optimal dose of running for longevity. Is more
better or worse? Journal of the American College of Cardiology 65, 420–
422.
Leonard, D., Van Long, N., 1992. Optimal Control Theory and Static Optimization in
Economics. Cambridge University Press, Cambridge.
Liljas, B., 1998. The demand for health with uncertainty and insurance. Journal of
Health Economics 17, 153–170.
Liljas, B., 2000. Insurance and imperfect financial markets in Grossman’s demand
for health model – a reply to Tabata and Ohkusa. Journal of Health Economics
19, 811–820.
Liljas, B., 2002. An exploratory study on the demand for health, life-time income,
and imperfect financial markets. In: Lindgren, B. (Ed.), Individual Decisions for
Health. Routledge, London, pp. 29–40.
Locke, S., 1999. Exercise-related chronic lower leg pain. Australian Family Physician
28, 569–573.
Morton, J.P., Kayani, A.C., McArdle, A., Drust, B., 2009. The exercise-induced stress
response of skeletal muscle, with specific emphasis on humans. Sports Medicine
39, 643–662.
Must, A., Spadano, J., Coakley, E.H., Field, A.E., Colditz, G., Dietz, W.H., 1999. The disease
burden associated with overweight and obesity. JAMA: The Journal of the
American Medical Association 282, 1523–1529.
Muurinen, J.M., 1982. Demand for health. A generalised Grossman model. Journal
of Health Economics 1, 5–28.
Muurinen, J.M., Le Grand, J., 1985. The economic analysis of inequalities in health.
Social Science & Medicine 20, 1029–1035.
Oniki, H., 1973. Comparative dynamics (sensitivity analysis) in optimal control theory.
Journal of Economic Theory 6, 265–283.
Picone, G., Uribe, M., Wilson, M.R., 1998. The effect of uncertainty on the demand
for medical care, health capital and wealth. Journal of Health Economics 17,
171–185.
Randolph, C., 2008. Exercise-induced bronchospasm in children. Clinical Review of
Allergy and Immunology 34, 205–216.
Sabanayagam, C., Shankar, A., 2010. Sleep duration and cardiovascular disease: results
from the National Health Interview Survey. Sleep 33, 1037–1042.
26
K. Bolin, B. Lindgren / Journal of Health Economics 50 (2016) 9–26
Schnohr, P., O’Keefe, J.H., Marott, J.L., Lange, P., Jensen, G.B., 2015. Dose of jogging
and long-term mortality. The Copenhagen city heart study. Journal of the
American College of Cardiology 65, 411–419.
Selden, T.M., 1993. Uncertainty and health care spending by the poor: the health
capital model revisited. Journal of Health Economics 12, 109–115.
Shone, R., 2002. Economic Dynamics – Phase Diagrams and Their Economic
Application. Cambridge University Press, Cambridge.
Steinhausen, H.C., 2002. The outcome of anorexia nervosa in the 20th century.
American Journal of Psychiatry 159, 1284–1293.
Steinhausen, H.C., Weber, S., 2009. The outcome of bulimia nervosa: findings from
one-quarter century of research. American Journal of Psychiatry 166, 1331–1341.
Swedish National Food Agency, 2015. Nordic Nutrition Recommendations, 2012.
www.slv.se.
Tisi, P.V., Shearman, C.P., 1998. The evidence for exercise-induced inflammation in
intermittent claudication: should we encourage patients to stop walking?
European Journal of Vascular and Endovascular Surgery 15, 7–17.
Vineis, P., Alavanja, M., Buffler, P., Fibtgan, E., Franceschi, S., Gao, T., Gupta, P.C.,
Hackshaw, A., Matos, E., Samet, J., Sitas, F., Smith, J., Stayner, L., Straif, K., Thun,
M.J., Wichmann, H.E., Wu, A.H., Zaridze, D., Peto, R., Doll, R., 2004. Tobacco and
cancer: recent epidemiological evidence. Journal of the National Cancer Institute
96, 99–106.