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Journal ArticleDOI

A Closed-Form Solution for the Health Capital Model

TL;DR: It is shown that the optimal solution always implies eternal life and this outcome occurs independently from the initial stock of health, the impact of health on productivity, and the importance of health for utility.

AbstractThis paper provides a closed-form solution for the health capital model of health demand. The results are exploited in order to prove analytically the comparative dynamics of the model. Results are derived for the so called pure investment model, the pure consumption model and a combination of both types of models. Given the plausible assumptions that (i) health declines with age and that (ii) the health capital stock at death is lower than the health capital stock needed for eternal life, it is shown that the optimal solution always implies eternal life. This outcome occurs independently from the initial stock of health, the impact of health on productivity, and the importance of health for utility and it is robust against the introduction of a finite age-dependent rate of health depreciation.

Summary (2 min read)

1. Introduction

  • This paper provides for the first time a closed-form solution of the health capital model of health demand.
  • In any period or, in continuous time, in any instant of time, health capital depreciates and is potentially augmented by health investment.
  • After acknowledging that a finite life requires that the fixed point for health capital lies below the minimum health needed for survival, the paper continues without debating the potential logical inconsistency involved in this assumption.

2. The Model

  • In order to derive a closed-form solution the authors need to assume that the utility function and the production function are iso-elastic.
  • The authors assume that goods consumption provides always utility and that health may or may not enter the utility function, 0 < β ≤ 1.
  • These modifications would not change the basic mechanics of the model because the first order conditions are structurally identical in both cases.
  • The original Grossman model additionally assumes that the production of health needs also a time input beyond health expenditure.

3. The Solution

  • Given (14), which is assumed until Section 5, individuals prefer a constant consumption share and thus a constant share of health care expenditure throughout their life.the authors.
  • This means that the explicit solution does not require an implausible assumption about the value of σ.
  • In other words, the optimal solution remains interior when the rate of health depreciation increases.

4. Comparative Dynamics

  • Initially healthier people are healthier at any given age t.
  • As individuals age their health capital stock is declining if their initial health is larger than H∗ and rising if their initial health is lower than H∗. Proposition 5 (Income and Medical Technology).
  • Irrespective of the power of medical technology, the weight of health in utility, and income, eternal life is the optimal solution and it is approached from everywhere, i.e. for any state of initial health.
  • Notice from (18) that H∗ is approached from any initial condition.
  • The available discussion of the comparative statics of the Grossman model has focussed on models with constant δ.

5. Generalization

  • Since a closed-form solution exists only for a special parametrization the question naturally occurs how general these results are.
  • The following proposition establishes that the qualitative features regarding the steady state of immortality are universal.
  • The special case and the general case share the same steady state.
  • The not-drawn special case where σ = σ̃, is reached when the ẋ = 0–isocline is horizontal and coincides with the stable saddlepath.
  • The expenditure share of health care increases with age and deteriorating health capital stock if and only if σ < σ̃.

6. Conclusion

  • This paper has provided an analytical closed-form solution of the Grossman model.
  • It exhibits a unique saddlepath-stable fixed point at which health does not deteriorate.
  • Global convergence towards immortality is a troubling prediction.
  • With the present paper at hand it is easy to see how it reverts the equilibrating forces of the Grossman model.
  • Because of its gerontological foundation the model of health deficit accumulation is straightforwardly calibrated with real data.

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Strulik, Holger
Working Paper
A closed-form solution for the health capital model
cege Discussion Papers, No. 222
Provided in Cooperation with:
Georg August University of Göttingen, cege - Center for European, Governance and Economic
Development Research
Suggested Citation: Strulik, Holger (2014) : A closed-form solution for the health capital model,
cege Discussion Papers, No. 222, University of Göttingen, Center for European, Governance
and Economic Development Research (cege), Göttingen
This Version is available at:
http://hdl.handle.net/10419/103879
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ISSN: 1439-2305
Number 222 – November 2014
A CLOSED-FORM SOLUTION FOR THE
HEALTH CAPITAL MODEL
Holger Strulik

A Closed-form Solution for the Health Capital Model
Holger Strulik
November 2014.
Abstract. This paper provides a closed-form solution for the health capital model
of health demand. The results are exploited in order to prove analytically the
comparative dynamics of the model. Results are deri ved for the so called pure
investme nt model, the pu r e consumption model and a combination of both types of
models. Given the plausible assumption s that (i) health declines with age and that
(ii) the health capital stock at death is lower than the health capital stock needed
for eternal life, it is shown that the optimal solution always impli e s et er n al li fe . This
outcome occurs independently from the initial st ock of health, the impact of health
on producti v i ty, and the importance of heal t h for utility and it is robust against the
introduction of a finite age-dependent rate of health depreciation.
Keywords: Longevity, Health, Health Care Demand.
JEL: D91, J17, J26, I12.
I would li ke to thank Carl-Johan Dalgaard, Ben Heijdra, Volker Meier, Gustav Feichtinger, Johannes
Schuenemann, and Timo Trimborn fo r d i sc u ss io n and helpful comments.
University of Goettingen , Department of Economics, Platz der Goettinger Sieben 3, 37073 Goettingen,
Germany; email: h o lg e r.st ru l ik@ wi wi .u n i- g oett in g en . d e.

1. Introduction
This paper provides for the first time a closed-form solution of the health capital model of
health demand. The model is also known as the Gr ossm an model, named after the seminal paper
of Grossman (1972), which develop ed its main ingredient s . In its long history the Grossm an
model has been criticized for various s hor t com i ngs and counterfactual predictions. Several of
these (alleged) shortcomings have been addressed by further developments of the original model.
The core mechanics of the Grossman model is the conventional paradigm in the economics of
health demand and remained, until recently, basically unchallenged by the development of an
alternative theory. Empirically, the Grossman model is the inspiration if not the foundation of
many redu ce d- for m and str uc t ur al models of health demand.
The core me chanics of the Grossman model arise from the assumption that individuals ac-
cumulate health c api t al H in a simi l ar fashion as they accumulate human capital in form of
education. In any period or , in continuous time, in any instant of ti me , health capit al depreci-
ates and is potentially augment ed by health investment. The health capital stock of an individual
of age t thus evolves, in continuous time, according to
˙
H(t) = f(I(t)) δ(t )H(t), in which I
is investment, f is a positive function, and δ is the depreciation rate. The key assumption is
that the loss of health capit al through depreciation is an increasing function of its stock. This
means that of two individuals of the same age t, the one in bett er health, i.e. the one with the
greater health stock H(t) loses more health capital in the next instant, since health depreciati on
δ(t)H(t) is increasing in H(t). Notice that this basic assumption is imposed independently from
whether δ is considered to be constant or age-dependent.
1
The notion of health capital accumulation according to the Grossman mode l c ontradicts basic
insights from modern gerontology. There, the human life course is understood as “intrinsic,
cumulative, progr es si ve, and deleterious loss of function that eventually culminates in death.”
(Arking, 2006, Masoro, 2006). Evidence from gerontology support s the reverse of the Grossman
assumption. The accu mulation of health deficits is foun d to be a positive fun ct i on of the health
deficits that are already present in an individual. Of two individuals of the same age the
unhealthier one is predicted to los e more heal t h (accumulate more health deficits) in the next
instant. This law of health deficit accumulation has a micro-fou nd at i on in reliability th eor y and
1
This paper is not the first one that observes this potentially problematic assumption of the Grossman model,
see, for example, Case and Deaton (2005), McFadden (2008).
1

it is a ver y stron g predi ct or of mortal i ty (M i tn i t sk i et al., 2002a, 2002b, 2005, 2006).
In defense of the Grossman model one could argue, based on Friedman ( 1953) , that a theory’s
assumptions should not matter as long as its pre di ct i ve quality is good. Generating test ab l e
predictions f rom the Grossman model, however, is a tough task. In order to appreciate this
fact, notic e that even the simplest ve r si on of the Grossman model generates two differential
equations (or in discrete t i me two difference equations): one equation of motion for the health
capital stock and one equation of motion generated from the first order conditions for optimal
health investment. The latter could be expressed as equation of motion for the shadow price
of health, or health investment, or consumption. The solution is thus expressed as a trajectory
in a two-dimensi on al phas e spac e. The problem is that there are infinitely m any trajectories
fulfilling the first order conditions, us ual l y pointing in all possible directions in the phase space.
In other words, based solely on the first order condi t i ons and th e equat i on of moti on for the
state variable (i.e. health capital), the solution is indeter mi nat e . The unique optimal solut i on of
the Grossman model is identified by the transversality condition. This unique optimal solution
allows to deri ve test abl e pred i ct i on s of the model.
It is perhaps fair to say that most of the problems that the literature had with solving the
Grossman model originated from an inappropriate use of the tr an sversality condition. Grossman
(1972) and some followers (e.g. Jacobsen, 2000) just ignored the transversality condition, others
had problems of applying it appropriately because they stated the health demand problem in
discrete time (Ried, 1998). Neglecting the tr an sversality condition is particul arl y worrying when
reduced-form or structural equat ion s for empirical estimation are deri ved. Many applications
derive these equations for health care demand from solving simplified ve rs i ons of the first order
conditions and the equation motion (Muurinen, 1982; Wagstaff, 1986; Grossman, 2000). But
since there are infinitely many trajectories fulfilling the r st order conditions, any structural form
obtained by ignoring the transversality condition is a result from (unwarranted) simplifications.
2
Some other studies suggested to reformulate the original Grossman model in order to reduce
the difficulties involved with identification. The original Grossman model assumes that death is
2
For exa m p le, Muurinen (1982 ) assumes that
˙
H/H is constant, i.e. an expon ential decline (or increase) of
health with age is assumed rather than derived. Muurinen actually states the transversality condition but then
ignores it in the derivation of health care demand. Similarly, Wagstaff (1986) accurately states a problem of free
terminal t i me but never invokes the transversality condition when solving for the structural form. Instead h e
records carefully the steps of simpli fyi n g assumption which distil from the infinitely many solution of the first
order conditions one particular set of estimation equations.
2

Citations
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Journal ArticleDOI
TL;DR: A novel approach to gauge the extent to which gender differences in longevity can be attributed to gender-specific preferences and health behavior and offers also an economic explanation for why the gender gap declines with rising income.
Abstract: In developed countries, women are expected to live about 4-5 years longer than men. In this paper, we develop a novel approach to gauge the extent to which gender differences in longevity can be attributed to gender-specific preferences and health behavior. We set up a physiologically founded model of health deficit accumulation and calibrate it using recent insights from gerontology. From fitting life cycle health expenditure and life expectancy, we obtain estimates of the gender-specific preference parameters. We then perform the counterfactual experiment of endowing women with the preferences of men. In our benchmark scenario, this reduces the gender gap in life expectancy from 4.6 to 1.4 years. When we add gender-specific preferences for unhealthy consumption, the model can motivate up to 89 percent of the gender gap. Our theory offers also an economic explanation for why the gender gap declines with rising income.

46 citations


Cites background from "A Closed-Form Solution for the Heal..."

  • ...As shown in Strulik (2015b) and Schünemann et al. (2017), allowing death to be a stochastic event adds more realism and complexity but contributes very little to the understanding of mechanisms and leaves quantitative results virtually unchanged....

    [...]

  • ...For a general critique of the health capital model, see Zweifel and Breyer, 1997; Case and Deaton, 2005; Almond and urrie, 2011; Dalgaard and Strulik, 2015; Strulik, 2015a).2 An exception is the study by Felder (2006), which avoids the roblematic health capital assumption and shows that a gender ap…...

    [...]

  • ...…critique of the health capital model, see Zweifel and Breyer, 1997; Case and Deaton, 2005; Almond and urrie, 2011; Dalgaard and Strulik, 2015; Strulik, 2015a).2 An exception is the study by Felder (2006), which avoids the roblematic health capital assumption and shows that a gender ap in…...

    [...]

  • ...(6) reduces to the health Euler equation derived in Dalgaard and Strulik (2014). For ̨ > 0, the health component of the utility function reduces growth of health investments over the life cycle....

    [...]


Posted Content
Abstract: We develop a life cycle model featuring an optimal retirement decision in the presence of physiological aging. In modeling the aging process we draw on recent advances within the fields of biology and medicine. In the model individuals decide on optimal consumption during life, the age of retirement, and (via health investments) the timing of their death. Accordingly, "years in retirement" is fully endogenously determined. Using the model we can account for the evolution of age of retirement and longevity across cohorts born between 1850 and 1940 in the US. Our analysis indicates that 2/3 of the observed increase in longevity can be accounted for by wage growth, whereas the driver behind the observed rising age of retirement appears to have been technological change in health care. Both technology and income contribute to the rise in years in retirement, but the contribution from income is slightly greater.

33 citations


Journal ArticleDOI
Abstract: This study presents a new view on the association between education and longevity. In contrast to the earlier literature, which focused on inefficient health behavior of the less educated, we investigate the extent to which the education gradient can be explained by fully rational and efficient behavior of all social strata. Specifically, we consider a life-cycle model in which the loss of body functionality, which eventually leads to death, can be accelerated by unhealthy behavior and delayed through health expenditure. Individuals are heterogeneous with respect to their return to education. The proposed theory rationalizes why individuals equipped with a higher return to education chose more education as well as a healthier lifestyle. When calibrated for the average male US citizen, the model motivates about 50% percent of the observable education gradient by idiosyncratic returns to education, with causality running from education to longevity. The theory also explains why compulsory schooling has comparatively small effects on longevity and why the gradient gets larger over time through improvements in medical technology.

29 citations


Cites background from "A Closed-Form Solution for the Heal..."

  • ...5 Strulik (2015b) investigates a stochastic version of the basic model of optimal aging by Dalgaard and Strulik (2014) and shows that the quantitative predictions are robust against the consideration of death as a stochastic event. human functioning (Gavrilov and Gavrilova, 1991)....

    [...]

  • ...For example, without further amendments, the health capital model predicts eternal life (Case and Deaton, 2005; Strulik, 2015a) and when death is enforced, the model usually predicts that health investments decline in old age and near death (Wagstaff, 1986; Zweifel and Breyer, 1997; Strulik, 2015a)....

    [...]


Journal ArticleDOI
TL;DR: A theory of aging and longevity according to which the probability to survive at any age depends on the number of health deficits accumulated, as suggested by modern biology is proposed.
Abstract: This paper proposes a theory of aging and longevity according to which the probability to survive at any age depends on the number of health deficits accumulated, as suggested by modern biology. I calibrate the model for a reference U.S. citizen and exploit the fact that the association between health deficits and mortality has been estimated with great precision in gerontology. This allows me to use the model for quantitative experiments. I compare health expenditure, health deficits, and the value of life for richer and poorer individuals. The model motivates a strong socioeconomic gradient of health. It explains how income growth and advances in medical technology and the induced increase in health care demand led to large reductions in health deficits and roughly a doubling of the value of life at all ages.

27 citations


Journal ArticleDOI
TL;DR: This paper sets up a a life cycle model in which individuals are subject to physiological aging, calibrate it with data from gerontology, and compare behavior and outcomes of adapting and non-adapting individuals.
Abstract: Aging humans adapt to their worsening state of health and old people are usually happier than estimated by young individuals. In this paper we investigate how adaptation to a deteriorating state of health affects health spending, life expectancy, and the value of life. We set up a a life cycle model in which individuals are subject to physiological aging, calibrate it with data from gerontology, and compare behavior and outcomes of adapting and non-adapting individuals. While adaptation generally increases the value of life (by about 2 to 5 percent), its impact on health behavior and longevity depends crucially on whether individuals are aware of their adaptive behavior.

18 citations


Cites background or methods or result from "A Closed-Form Solution for the Heal..."

  • ...Overall we confirm the observation of Strulik (2015b), made in the context of no adaptation, that including uncertain survival adds more realism (and complexity) but changes outcomes and predictions only marginally....

    [...]

  • ...…eternal life (Case and Deaton, 2005; Strulik, 2015a) and when death is enforced by design, the Grossman model usually predicts, counterfactually, that health investments decline in old age and near death (Wagstaff, 1986; Zweifel and Breyer, 1997; Strulik, 2015a; Dalgaard and Strulik, 2014b)....

    [...]

  • ...1Earlier quantitative studies using the health deficit model were concerned with the Preston curve (Dalgaard and Strulik, 2014a), the education gradient (Strulik, 2015), and the long-term evolution of the age at retirement (Dalgaard and Strulik, 2012)....

    [...]

  • ...For example, without further amendments, the Grossman model predicts eternal life (Case and Deaton, 2005; Strulik, 2015a) and when death is enforced by design, the Grossman model usually predicts, counterfactually, that health investments decline in old age and near death (Wagstaff, 1986; Zweifel…...

    [...]

  • ...5This way deteriorating health may motivate a hump-shaped age-profile of consumption, see Strulik (2015c)....

    [...]


References
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TL;DR: A model of the demand for the commodity "good health" is constructed and it is shown that the shadow price rises with age if the rate of depreciation on the stock of health rises over the life cycle and falls with education if more educated people are more efficient producers of health.
Abstract: The aim of this study is to construct a model of the demand for the commodity "good health." The central proposition of the model is that health can be viewed as a durable capital stock that produces an output of healthy time. It is assumed that individuals inherit an initial stock of health that depreciates with age and can be increased by investment. In this framework, the "shadow price" of health depends on many other variables besides the price of medical care. It is shown that the shadow price rises with age if the rate of depreciation on the stock of health rises over the life cycle and falls with education if more educated people are more efficient producers of health. Of particular importance is the conclusion that, under certain conditions, an increase in the shadow price may simultaneously reduce the quantity of health demanded and increase the quantity of medical care demanded.

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Abstract: This chapter contains a detailed treatment of the human capital model of the demand for health which was originally developed in 1972. Theoretical predictions are discussed, and theoretical extensions of the model are reviewed. Empirical research that tests the predictions of the model or studies causality between years of formal schooling completed and good health is surveyed. The model views health as a durable capital stock that yields an output of healthy time. Individuals inherit an initial amount of this stock that depreciates with age and can be increased by investment. The household production function model of consumer behavior is employed to account for the gap between health as an output and medical care as one of many inputs into its production. In this framework the “shadow price” of health depends on many variables besides the price of medical care. It is shown that the shadow price rises with age if the rate of depreciation on the stock of health rises over the life cycle and falls with education (years of formal schooling completed) if more educated people are more efficient producers of health. An important result is that, under certain conditions, an increase in the shadow price may simultaneously reduce the quantity of health demanded and increase the quantities of health inputs demanded.

904 citations


"A Closed-Form Solution for the Heal..." refers background in this paper

  • ...Many applications derive these equations for health care demand from solving simplified versions of the first order conditions and the equation motion (Muurinen, 1982; Wagstaff, 1986; Grossman, 2000)....

    [...]


Journal ArticleDOI
Abstract: Over the past half century, Americans spent a rising share of total economic resources on health and enjoyed substantially longer lives as a result. Debate on health policy often focuses on limiting the growth of health spending. We investigate an issue central to this debate: Is the growth of health spending a rational response to changing economic conditions—notably the growth of income per person? We develop a model based on standard economic assumptions and argue that this is indeed the case. Standard preferences— of the kind used widely in economics to study consumption, asset pricing, and labor supply—imply that health spending is a superior good with an income elasticity well above one. As people get richer and consumption rises, the marginal utility of consumption falls rapidly. Spending on health to extend life allows individuals to purchase additional periods of utility. The marginal utility of life extension does not decline. As a result, the optimal composition of total spending shifts toward health, and the health share grows along with income. In projections based on the quantitative analysis of our model, the optimal health share of spending seems likely to exceed 30 percent by the middle of the century.

759 citations


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This paper provides a closed-form solution for the health capital model of health demand.