# 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 (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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##### Citations

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....

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...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…...

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...…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…...

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...(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....

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33 citations

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)....

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...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)....

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27 citations

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....

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...…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)....

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...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)....

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...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…...

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

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##### References

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### "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)....

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759 citations