A Closed-Form Solution for the Health Capital Model
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
18 citations
Cites background or methods from "A Closed-Form Solution for the Heal..."
...It has been shown elsewhere (Strulik, 2015a; Schuenemann et al., 2017a) that treating death as a health-dependent stochastic event adds more complexity but only marginally changes the predictions derived from the health deficit model....
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...2 For a critique of the health capital model, see also Wagstaff, 1986; Zweifel and Breyer, 1997; Case and Deaton, 2005; Almond and Currie, 2011; Dalgaard and Strulik (2015), and Strulik (2015b)....
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...An infinite life is actually supported by Grossman’s (1972) health capital model, which exhibits a steady state of constant health (see Strulik, 2015b)....
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...In contrast to the health capital model, the health deficit model, as calibrated in Section 3, exhibits no steady state (see Dalgaard and Strulik, 2015, for a discussion of steady states in the health deficit model)....
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16 citations
13 citations
Cites background or methods from "A Closed-Form Solution for the Heal..."
...As shown in Strulik (2015a) and Schuenemann et al. (2017b), stochastic versions of the health deficit model in which the probability of death depends positively on health deficits add more realism at the price of higher complexity but provide little extra insight into life-cycle choices and…...
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...We solve model by applying the relaxation algorithm of Trimborn et al. (2008). We begin with the life cycle choices of the non-depressed benchmark American, which are shown in Figure 1 by blue (solid) lines....
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...…the standard Euler equation for consumption adjusted by health deficits, Intuitively, the accumulation of health deficits drives down the incentive to spend on consumption in old age, and thus lowers the slope of the lifetime consumption path (which may become hump-shaped; see Strulik, 2015b)....
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5 citations
Cites background from "A Closed-Form Solution for the Heal..."
...In the standard model of health capital accumulation (Grossman, 1972) there always exists a steady state of constant health such that individuals inevitably live forever (Strulik, 2015)....
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...In the health deficit model (Dalgaard and Strulik, 2014, 2015), a steady state of constant health exists as well, but only for a favorable constellation of parameters....
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4 citations
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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822 citations