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.

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 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

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.

Figures

Figure 1: Aging According to the Grossman Model

Figure 2: Phase Diagrams: Left: σ < σ̃. Right: σ > σ̃

Figure A.1: The Curve f(H)

Full text

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 fi 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

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