## Abstract

### Aim

By analyzing the Russian demand for medical care, we aim to highlight the practical relevance of Grossman’s health investment model, one of the most important developments in health economic theory.

### Subjects and methods

We refer to the often-criticized inconsistencies between the health investment model’s theoretical implications and its empirical results. While the theoretical model implies, inter alia, that the demand for medical care increases with improving health, empirical tests show conflicting results. In order to solve this inconsistency, we specify the model’s inherent health investment production function to be of decreasing rather than constant returns to scale. Afterwards, we employ our derived structural demand function for medical care in an empirical panel analysis using data from the Russia Longitudinal Monitoring Survey for 1996–2008.

### Results

In contrast to other studies, we show that even Grossman’s standard model setting generates a structural demand function for medical care that implies sick people use more medical care, provided that the functional form of the health investment production function is properly specified. Further, our empirical analysis affirms our theoretically derived implications and provides important new insights into the Russian demand for medical care.

### Conclusions

Grossman’s health investment model is of practical relevance to health economists.

## Notes

Instrumenting health with childhood health or a binary indicator of one parent’s smoking during childhood Galama et al. (2012) estimate insignificant health coefficients. Nevertheless, their individual health expenditure data have been imputed and hence may include potential measurement errors (see, Galama et al. 2012). Moreover, there are possible concerns regarding the accuracy of the retrospectively recalled childhood health data. Finally, considering strong associations between childhood health as well as childhood and adult socioeconomic status (see, e.g., Gilman 2002; Currie and Goodman 2010), the coefficients of their employed socioeconomic variables in the generally less efficient IV regressions might mitigate the estimation of the instrumented health coefficient because of severe problems of multicollinearity and endogeneity.

Galama (2011) continues to specify the health investment production function to be of constant returns to scale, but assumes the appreciation of health capital (noted in his study as the “health production process”) in the motion of health to be of decreasing returns to scale. At higher levels of health investment, very expensive treatments often provide only relatively small improvements in health (Galama et al. 2012). While this definition is possibly appropriate in cost-benefit analyses, it is rather misleading in the sense of investment models. In Grossman’s health investment model, the improvement in health is already generated by the health investment production function, which is by definition the gross increase in health stock.

For a discussion of the term structural demand for medical care, see Galama et al. (2012).

Although we assume a fixed endpoint problem instead of a fixed terminal time

*T*, we do not solve our model for optimal longevity, i.e., we do not consider life cycle patterns of health capital derived by the optimal adjoint paths considering the distinct transversality conditions. Based on our research question, we are primarily interested in the model’s flow and stock equilibriums given by Eqs. (11) and (12), which equal equation (8’) and (13) of Ehrlich and Chuma (1990).If constant returns to scale are assumed, the cost function of health investments is linear in output

*I(t)*, i.e., \( \frac{\partial^2{C}_I(t)}{\partial I{(t)}^2}=0 \). Therefore, Eq. (11) changes to \( {\pi}_H(t)=\eta (t) \), which is independent of*I*(*t*) (see Ehrlich and Chuma 1990). For comparison, see Eq. (31)If a constant returns to scale technology is assumed, Grossman’s familiar rule for the optimal stock of health capital can be derived as \( {\pi}_H(t)\left[\delta (t)+r-\frac{{\dot{\pi}}_H(t)}{\pi_H(t)}\right]=\frac{1}{\varphi_A(0)}\;{e}^{\left(r-\rho \right)t}\frac{\partial U(t)}{\partial h(t)}\frac{\partial h(t)}{\partial H(t)}+w\frac{\partial h(t)}{\partial H(t)} \). Here, the optimal amount of health investments is selected as a singular control in order to maintain this stock equilibrium as long as feasible. That is, consumers adjust to their desired stocks of health capital instantaneously, given the independence of \( {\pi}_H(t) \) and \( I(t) \) (Grossman 1972b). Given this identity of current and desired stock of health capital, an increase in the demand for health directly increases the demand for health investments and consequently the demand for medical services (Grossman 2000).

However, merging the more than 20 different household and individual data sets on different survey topics each year shows that a significant portion of households (and individuals) has not been surveyed regarding all of the survey topics (e.g., the data set with household income data includes only 3,560 households in 1996). Therefore, after an additional deletion of severe outliers, the number of total observations is reduced to a not balanced data set with maximally 81,273 observations for the merged data set of 11 waves.

The RLMS-HSE data set includes only variables exploring the question of whether individuals demand several types of medical care but not by how much they demand medical care. However, even if this information would be provided, such data fail to capture the intensity of services by duration of visits or the number of diagnostic tests performed, a general problem of studies on the demand for medical care (Wagstaff 2002).

In Russia, the public provision of health care predominates private financing. In this line, only 0.5 % of the observed Russians in the RLMS-HSE data set made direct payments for their hospital stay. Nevertheless, 17.54 % of all RLMS-HSE participants paid for their prescribed medicine (including informal payments).

The reported health status may be biased and therefore may inappropriately measure the latent variable overall health. Nevertheless, even with an overall health index, measurement problems of the latent construct possibly occur since the selection of necessary health problems and determination of their weights for index construction are rather subjective. Further, other health measures may entail a response bias by more complicated interviewer questions (Gerdtham et al. 1999).

Endogeneity problems arise from the interdependence of health capital and demand for medical care: demand for medical care depends on the actual health status and actual health is likely to be partly determined by prior demand for medical care.

It has to be noted that RLMS-HSE reported wage rate data are less informative because of wage arrears and delayed wage payment (Desai and Idson 2000). Therefore, the RLMS-HSE constructed real household income variable considers different wage variables and imputes missing values according to regional, gender, and age-specific relations.

Because our expenditure data have many zero observations, we further estimated the Poisson and Poisson pseudo-maximum-likelihood estimator (see Santos Silva and Tenreyro 2006). Coefficients are comparable to those presented here with a negative coefficient of health on the out-of-pocket expenditures in logs.

The Vuong test between the Poisson and a zero-inflated Poisson model regarding the number of hospital nights indicates that the zero-inflated model is not better than an ordinary Poisson regression model. Yet the efficiency of the fixed effects Poisson model is assumed to be relatively low since this estimation model considers only individuals with observations in all waves and additionally loses information of observations that have zero hospital nights in all waves (57,156 observations).

By contrast, Galama et al. (2012) note that exploiting the panel nature of their data with fixed effects estimators would result in insignificant coefficients of lagged health. Additionally, their instrumental variable approach also shows insignificant results. Nevertheless, these authors argue that contemporaneously rather than lagged health variables may be more appropriate since medical treatment likely responds quickly to changes in health.

Furthermore, higher levels of education are often associated with lower subjective discount rates and hence a higher subjective value of longevity (Ehrlich and Chuma 1990).

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## Acknowledgments

We would like to thank Titus Galama for his valuable suggestions. We would also like to express our gratitude for the fruitful discussion at The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid, Spain, from 7–11 July 2014.

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This article does not contain any studies with human participants or animals performed by any of the authors.

## Appendices

### Appendix A1: Derivation of equilibrium conditions

Applying the maximum principle (Pontryagin et al. 1967) to problem (10), the Hamiltonian function is defined for \( t\in \left[0,T\right] \) as

The Hamiltonian is jointly concave in both the state and control variables because of the strict concavity of the utility function, production of health, and the generation of healthy time (Ehrlich and Chuma 1990). For the optimality conditions, the Hamiltonian system of equations is given by:

and

with \( {\varphi}_H(T)={\varphi}_H\ge 0,\;H(T)\le {H}_{min},\;\frac{\partial \mathcal{H}(T)}{\partial T}<0, \)

It follows from (16)-(22) that the adjoint variables are positive and health investments are non-negative given a production function with decreasing returns to scale. From (19), the flow equilibrium condition for \( t\in \left[0,T\right] \) can be derived as

From (20), which is a backward ordinary differential equation, it follows by some rearrangements that

and from (24) it follows that

Substituting (20) and (24)-(26) in (21), a continuous stock equilibrium condition for health can be derived as follows

with

### Appendix A2: Structural demand for medical care

Assuming non-joint production functions, the consumer produces health investments and commodities by

with \( \kappa +\mu <1 \), \( \alpha =\frac{1}{\kappa +\mu } \), and \( \zeta +\vartheta =1 \).

The constants κ and μ are the output elasticities of market inputs and time for health investments, and the constants *ζ* and *ϑ* are the output elasticities of market inputs and time for commodity productions, respectively. The parameter α stands for the inverse scale elasticity. Because of decreasing returns to scale, \( \frac{1}{\alpha }<1 \) and therefore \( \alpha >1 \).

Using the Lagrange multiplier technique, the dual cost function for an investment in health takes the form

with \( {\pi}_H(t)=\left[{\left(\frac{\kappa }{\mu}\right)}^{\frac{\mu }{\kappa +\mu }}+{\left(\frac{\kappa }{\mu}\right)}^{-\frac{\kappa }{\kappa +\mu }}\right]{p}_M{(t)}^{\frac{\kappa }{\kappa +\mu }}\;w{(t)}^{\frac{\mu }{\kappa +\mu }}\;E{(t)}^{-\frac{1}{\kappa +\mu }} \).

Further, the minimum cost of producing a given output of *Z(t)* is given by

with \( {\pi}_Z(t)=\left[{\left(\frac{\zeta }{\vartheta}\right)}^{\vartheta }+{\left(\frac{\zeta }{\vartheta}\right)}^{-\zeta}\right]{p}_Q{(t)}^{\zeta}\;w{(t)}^{\vartheta }E{(t)}^{-1} \).

The resulting cost-minimizing factor inputs of the health investment production are of the following forms:

and

Solving (23) for (31) and writing it in the double logarithmic form yields

with \( {\beta}_{01}= \ln \left[{\left(\frac{\kappa }{\mu}\right)}^{\frac{\mu }{\kappa +\mu }}+{\left(\frac{\kappa }{\mu}\right)}^{-\frac{\kappa }{\kappa +\mu }}\right] \) and \( \kappa ={\beta}_5 \).

Further, Grossman (2000) approximates the production function of healthy time by the functional form

with \( {\beta}_1>0\;\mathrm{and}\;{\beta}_2>0 \).

It follows from Eq. (36) that \( \frac{\partial {h}_i}{\partial {H}_i}>0 \) and \( \frac{\partial^2{h}_i}{\partial {H_i}^2}<0 \), i.e., the marginal productivity of health capital regarding the production of healthy time is assumed to be positive but decreasing. According to Wagstaff (1986), the approximated depreciation rate function takes on the form

with \( {\beta}_3>0 \) indicating an increasing depreciation rate as the individual ages. Additionally, the components of vector **β**
_{
4
} will be positive if the respective individual characteristics of vector **X**
*(t)* are damaging to health.

Finally, we have to approximate \( \ln\;{\psi}_1(t) \). Grossman (2000) assumes \( r(t)-\frac{\dot{\pi}(t)}{\pi (t)}=0 \) and hence \( {\psi}_1=1 \). Wagstaff (1986) assumes \( r(t) \) and \( \frac{\dot{\pi}(t)}{\pi (t)} \) to be constant and therefore \( \dot{\psi}(t)>0 \) with \( \ln {\psi}_{1i}(t)={\beta}_{9\;}{t}_i \) because \( \dot{\delta}(t)>0 \). Considering the life path of the relative shadow price of health (28), \( \eta (t) \) depends on \( \eta (T) \) and the subjective value of healthy life, i.e., the discounted value of health benefits accruing over the remaining life span. Following the argumentation of Ehrlich and Chuma (1990), we assume \( \dot{\eta}(t)>0 \) dictated by a generally declining health-age profile. Additionally, the life path of the relative shadow price of health is a function of wealth. In conclusion, we approximate \( {\psi}_1 \)
*(•)* by

The signs of \( {\beta}_7,\;{\beta}_9 \) and \( {\beta}_{10} \) are assumed to be positive since initial wealth \( {A}_i(0) \) and permanent income \( \sum_{t=t^{*}}^{T^{*}} ln{w}_i(t) \) increase lifetime wealth, which are both assumed to generally raise the value of life extension. Age \( {t}^{*} \) is the age at earning wages for the first time and age \( {T}^{*} \) is the age at retirement.

Substituting (35)–(38) in the logarithmic stock equilibrium condition (13) of the pure investment model gives

with \( {u}_{1i}=-\frac{1}{\alpha -1} \ln {\delta}_0 \) and \( {\beta}_{02}=\frac{ln{\beta}_1{\beta}_2-{\beta}_{01}-ln\alpha }{\alpha -1}. \)

Substituting (39) in the logarithmic form of the cost-minimizing factor input for medical care (33) results in

with \( {\beta}_{03}=\left(1-{\beta}_5\alpha \right) \ln \left(\frac{\beta_5}{\frac{1}{\alpha }-{\beta}_5}\right)+\alpha {\beta}_{02} \) and \( {u}_{2i}={\alpha u}_{1i} \).

### Appendix A3: Empirical analysis

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Burggraf, C., Glauben, T. & Grecksch, W. New impacts of Grossman’s health investment model and the Russian demand for medical care.
*J Public Health* **24**, 41–56 (2016). https://doi.org/10.1007/s10389-015-0692-5

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DOI: https://doi.org/10.1007/s10389-015-0692-5