Estimating a Health Production Function for Brazil: Some New Evidence

This paper reports the impact of low-cost health centers on child mortality in Brazil. We use a comprehensive database to evaluate the impact of a change in Brazilian health policy from 2006 to 2009, when the number of health centers per capita increased significantly while hospitals per capita were reduced, indicating a focus on low-cost, low-complexity medical services. Unlike most empirical studies, our results indicate that additional health care decreases mortality. Increasing the number of health centers per capita decreases mortality through access to basic services and make high-complexity hospitals more effective, as they can focus on more serious conditions.


Introduction
Medical care expenditures have been rising rapidly in most countries. According to the World Bank (2020), total health expenditure in Brazil grew from 8.3% of Gross Domestic Product (GDP) in 2000 to 9.5% in 2017. This increase has come both from the private and the public sectors: public health expenditure increased from 3.5% in 2000 to 4% in 2017, accounting for more than 40% of total health expenditure. This upward trend is expected to continue, as most developed countries have even higher (and still increasing) percentages of GDP dedicated to health. For a comparison, in the United States, this share rose from 12.5% in 2000 to 16.4% in 2010 and 17.1% in 2017. In France, it went from 9.6% in 2000 to 11.3% in 2017; in Germany, in the same period, it increased from 9.8% to 11.2%. Theoretical Economics Letters A recurrent policy question is whether higher health expenditure actually improves the health status of the population. Several empirical studies present a negative answer. Thornton (2002) finds that medical care expenditures have no impact in American states, confirming earlier studies such as Auster et al. (1972).
In developing countries, evidence is even harsher: health care expenditure has a negative impact on life expectancy, since it may be squeezing out more valuable inputs such as food or water 1 . This paper revisits this issue by investigating the relationship between the age-specific mortality rate of children up to 5 years old and the availability of low-cost health centers. This rate is defined as the ratio between the number of deaths of individuals under five years old and the population of that age in each year. (Formally, it is not the same as the under-five mortality rate, which is the ratio between the number of deaths in this age range per 1.000 births, although both measures capture the death risk of under-five children.) The main reason to choose child mortality is that children are particularly sensitive to health care.
Moreover, lower child (and infant) mortality rates have significant impacts on a populations life expectancy. Generally, most concepts used to evaluate the benefits of medical interventions take into account how many years patients are expected to live after the intervention, highlighting the importance of child care (the concept of disability-adjusted life years 2 ). We restrict our attention to mortality due to diseases, as opposed, for example, to deaths due to violence or accidents.
With a child mortality rate of 20.6 per 1000 in 2009, Brazil performed poorly even when compared to countries with lower levels of income per capita (Colombia, for instance, had 19.7 in 2000) and to other Latin American countries (Mexico 17.6, Argentina 14.2 and Chile 8.9 in the same year), according to the World Bank 3 -see Table 1 for details. This is most surprising considering that Brazil spends more on health as a share of GDP than many countries with higher per capita income, such as Chile (8.2%), Russia (5.4%) and Mexico (6.5%)-all of which were able to attain much lower child mortality rates. Again, this suggests that lack income is not the main restriction in Brazil.
These figures strongly suggest misdirected health expenditures. Brazil had an underfunded health system for a long time. Also, the country invested in expensive hospitals instead of simpler centers able to treat minor conditions effectively and prevent them from becoming serious (and demanding expensive treatments). These factors resulted in an inefficient health structure.
In spite of these weak comparisons, both infant and child mortality rates in Brazil have been declining consistently. The age-specific mortality rate for the under-five population declined from 4.6 in 2002 to 3.7 in 2009 (per 1000 children at this age)-a reduction of 20%. 1 See Fayissa and Gutema (2005) for an estimation of a health production function in Sub-Saharan Africa.
2 See Murray (1994). 3 World Bank (2020). In our analysis, we were able to identify the impact of changes in the availability of health centers due to the fact that these changes were strongly diversified across states. While in some states it increased up to 40% in the four-year period under study, in others it decreased by as much as 25%, suggesting either different health policies were being implemented, or public expenditures on health were limited (an exceptional situation given the national trend discussed above).
We use health data for the period 2006 to 2009 from DataSus to evaluate the evolution of different health centers per capita in every state, building an aggregate measure of nearly-substitute centers. We then evaluate the impact of these changes on child mortality, one of the main indicators of child health. We choose this period for two main reasons. First, as mentioned, this substitution policy (hospitals for health centers) was particularly active. Second, due to the availability of data, as the measurement of some series we use was changed afterwards. We have two main results, discussed in detail in section 3: additional health centers per capita reduce child mortality (direct effect) and make hospitals more effective (indirect effect).
Although this is an empirical paper, there is a long tradition in the theoretical literature that provides foundations for it 4 . While we do not intend to cover that tradition here, it is worth mentioning that we are building on the concept of aggregate health production function, originally developed by Grossman (1972).
Individuals use different inputs to produce health: medical services, income, and education, among others. Notice that in order to apply this to child mortality, one must assume parents are making the best choices from the point of view of their children. Also, population-related characteristics refer to the whole of society, and exclude pre-school children themselves-who do not earn any income Theoretical Economics Letters or have any level of education. Hence we are controlling for the impact of adult features on infant health, not the impact of an individual's features on his or her own health, as suggested by aggregate health production functions. In short, we evaluate how these features affect the way parents take care of their children.
Lastly, although the focus of the paper is to study the impact of health structure on child mortality, it is well known that health policy is not the only policy that affects health outcome. In particular, educational and income policies are believed to have a large impact on health.
The BHS's full coverage is equivalent to complete insurance for public health: income is decreased through taxes and is reverted as free health services. Hence, a moral hazard problem arises: people have lower incentives to avoid risky behavior (ex-ante) and to take good care of an illness (ex-post). As a result, the BHS tends to amplify demand, as prices are not the restriction. The actual restriction comes from long lines and the unavoidable low quality of overused medical services (in fact, individuals also buy partial private insurance to overcome these issues). Hence, building a new health center is less effective than if people had more income but had to pay medical fees. For this reason, we interpret the results below as a lower bound on the effect of health centers on child mortality.
The remainder of the paper is organized as follows. In Section 2 we discuss our methodology. Section 3 presents the results. Section 4 concludes. The appendices collect all tables and provide a detailed description of the data.

Data and Methodology
We evaluate the impact of changes in the availability of public health centers on child mortality outcomes. We use annual aggregate data from DataSus and IBGE We use health units (per 1000 inhabitants, like all the other measures) instead of health expenditures because it allows us to make use of more detailed series: expenditures series are more aggregated. However, we are pooling together different types of institutions. We believe this should not create any major distortions as these institutions have similar capacities for medical interventions and, more importantly, can be viewed as nearly substitutes: one more unit will decrease the number of people that go to the other ones looking for same basic services provided in any of them. For the information on child mortality (0 -4 years), we used the information for general mortality by age also available in DataSus; and for population by age, from IBGE.
Our empirical strategy is to regress these state/year outcomes for child mortality on the number of health centers relative to the population of each state.
We ask the question: as the number of health centers per capita increases, does P. Hemsley, L. Hollanda DOI: 10.4236/tel.2020.105063 1078 Theoretical Economics Letters child mortality fall? As discussed in the previous sections, the dependent variable is the age-specific mortality rate of under-five children and the main regressor is the number of health centers per 1000 inhabitants. We include several controls, discussed below-see Table 2 for some descriptive statistics for the variables we used, which are detailed in the Appendix.
A first potential drawback for this empirical strategy is related to the direction of causality. On the one hand, we expect more health centers to decrease child mortality; on the other hand, public health policy is addressed primarily at states with high mortality rates, which should invert the expected sign of the estimated coefficient. While we tested a number of different instruments with similar results (so that the model was not oversensitive to the choice of instruments), we use the one-period lag of health centers per capita as our main instrument. First, it is highly correlated to the current value of health centers, as the series presents significant inertia. Second, it should not respond to future unexpected changes of child mortality. pregnant women and parents of newborns will use any health services available if they need to). Since most under-five deaths happen in the first year, we understand this is not a very restrictive assumption. The only delicate point for this assumption to hold is related to the health of pregnant women: if the health of a newborn or young child depends significantly on the health of the mother before the pregnancy, which in turn depends on the availability of health centers in the past, then some of our instruments may be weak. However, research indicates that the health of the mother during pregnancy is much more important to the health of the newborn than health before pregnancy (Gabbe et al., 2016). Nevertheless, if the use of lagged value is not completely effective at dealing with the endogeneity issue, the direction of the bias is positive (as higher mortality rates induce policymakers to increase the number of health centers); hence our estimators will be lower (in absolute value) than the true parameter and the probability of rejecting the null hypothesis will also be lower. In other words, our model underestimates the impact of health centers on child mortality. A similar reasoning applies to the other health-related variables in the model (discussed below): we assume the explanatory variables have no cumulative effect.
In order to avoid a possible problem caused by omitted variables, we bring into the model a series of control variables suggested by other empirical papers and by the theoretical literature. 5 The first control variable is income. As suggested by Grossman (1972), a higher income may be used to afford medical services and hence improve health outcomes. While in Brazil individuals cannot directly buy access to the public health system (due to the perfect-insurance structure of the BHS), they could potentially use their income to buy either private medical services, or other complementary goods and services related to improved health.
Although we do check different income measures, we focus on the proportion of poor in the population (defined as household per capita income lower than the poverty line, as discussed in the Appendix) because the impact of income is not linear: additional income improves health since individuals become able to afford significant inputs for health production (sanitation and nutrition are common examples). However, a higher income may be the result of work overload. The former effect should be dominant when income levels are very low, as demand is inelastic for basic services. Therefore we expect a positive sign: the higher the proportion of low income people in a society, the higher mortality rates should be.
Income also suffers from the same endogeneity problem as health services (Grossman, 2017): reversed causality. While more income may be used to improve health, better health also enables an individual to earn more income (say, higher productivity or more workhours). In relation to child mortality, this means parents of healthy children will have higher income. Again, the main instrument we use is the lagged value of income, which is strongly related to present income but cannot respond to present child mortality: accumulated income (from one year to the next) has no effect. We think this is reasonable for poor people, who hardly have any savings.
We also include the share of the population with private insurance as a control for access to private health services. We expect it to decrease mortality rates and to be correlated both with income and with the availability of public health structure. For the same reason our measure of health centers is endogenous, so should private insurance be: people tend to hire more insurance when they (or 5 Again, we reference the reader to Zweifel et al. (2009) for an overview and to Grossman (1972)  share as our instrument. We also include the share of the population with private insurance as a control for access to private health services. We expect it to decrease mortality rates and to be correlated both with income and with the availability of public health structure. For the same reason that our measure of health centers is endogenous, so should private insurance be: people tend to hire more insurance when they (or their children) have poor health. Analogously, we use the lagged value of this share as our instrument.
The third control is education. As suggested by numerous works, it has a large impact on health. Again, in spite of checking different measures, we focus on the proportion of the population with four or less years of formal schooling. Theoretical models suggest more educated societies are more efficient at health production (Grossman, 1972): they have more knowledge on prevention and treatments, parents are more suited to take care of infants and children. Notice also that most health-related knowledge is provided in basic education (it does not rise significantly whether one has 8 or 12 years of schooling). Hence we use the proportion of the population with four or less years of schooling as our measure of education. We expect a positive sign: the higher the proportion of uneducated people, the higher mortality rates should be.
It is not clear whether education also suffers from endogeneity. For example, the theoretical model of Grossman (1972) does not suggest it should be so. Moreover, we expect hospitals and health centers to have a strong interaction: the latter offers low-cost, low-complexity services, leaving room for the former to focus on high-complexity services. For this reason we also include an interaction term between hospitals and health centers.
One specific caveat comes from the type of physician made available at different points. While we still do not have conclusive results on it, we will mention it in section 4 below. We also include immunization coverage, as one of the main deterrents of major conditions. usually have a better health structure. 6 It is also related to the concentration of the population, which benefits from economies of scale in health services. Garbage collection is a measure of quality of sanitation services. We expect a negative sign, as sanitation is supposed to improve health (sanitation series have recently been discontinued, and we use direct garbage collection as a substitute.).
Lastly, notice that by choosing shares of the population we avoid dealing with possibly integrated series.
As mentioned in the introduction, health and educational policies do not act independently. First, the more educated people are, the more they will know how to use health services (and to follow received instructions). Second, efficient health services avoid family-disruptive events that prevent school-age children from attending school. In order to capture these effects, we include the interaction between health centers and education as a regressor.
Since we have panel data, one relevant point is the structure of potential unobserved fixed state effects. We do not assume them away. However, we assume they are uncorrelated with the time-varying regressors. In this case, a random-effect model is the most appropriate: fixed-effect estimators are inefficient and OLS test statistics will be incorrect. We will apply it throughout the next section, and come back to this topic in the last section.

Results
Our main results are reported, in terms of elasticities, in Notice initially that point estimates in Table 3 are systematically lower than in Table 4, reflecting the reverse causality problem: endogeneity biases the estimated coefficients of health centers and hospitals towards zero, since the impact of child mortality on health expenditures should be positive, while we expect both variables to have a negative coefficient. The coefficient of health centers, for example, is three times lower in Table 3, columns 2 (fixed effects) and 3 (random effects): in absolute value, they go from 0.7 to more than 2 -this is a large effect and suggests there is room for improvement considering the disparity of health centers per capita in our sample (it ranges from only 0.21 to 0.72). 7 We interpret this as evidence that results in Table 4 underestimate the actual impact of health centers and hospitals if the proposed instruments are not sufficient to eliminate endogeneity, as far as its main cause is reverse causality. We are then implicitly assuming that preferences for health are stable in short period of times such as our four-year span, as otherwise changes in preferences could be a source of endogeneity different from reversed causality.
Comparing the first three columns of   A preliminary point refers to the role of education. When a non-significant interaction term between education and health centers is added (column 5), the effect of health centers disappears; we cannot disentangle the effects of education and health centers then. The impact of the latter (through the interaction with hospitals alone) is reduced to 0.03.
Most of the controls had the expected sign, but were not significant in the main specification (Table 4, column 3). This has two explanations. First, time variation is low in the four-year span-highlighting the usefulness of the rapid expansion in health centers in this period. Second, some variables have low variation across states as a result of national policies. A noteworthy feature is related to the coefficient of income, which is not significant in any specification. We have two explanations for this. First, the BHS effectively provides medical insurance to the very poor-at least to the extent that it prevents child mortality.
Second, an income gain necessary to pull someone above the poverty line is not enough to give access to more sophisticated medical services. Another interpretation is that all the effect of income is captured through access to private insurance, which was quite significant.
Taking into account interaction effects, the total impact of health centers on child mortality varies over time, as the number of available hospitals change. In fact, this elasticity increases in absolute value, suggesting health policy has become more efficient. A similar point is valid for hospitals, which is particularly striking as their number was reduced.
In summary, from the results above, we believe the effect of health centers on child mortality can be split into two components, as was discussed in previous section. First, there is a direct impact. Second, it increases the efficiency of hospitals in treating more serious conditions.
For the sake of robustness, we note that the main results survive when we drop the lags and use current expenditures and general maintenance as instruments for health centers and hospitals. We have three general measures of public expenditures which we use as instruments for the variable health centers: Current expenditures (DC), General maintenance (DCus) and Total cost of government employee salaries (DCP). DCP is included in DCus, which is included in DC; for this reason we use the differences between them. These are general measures of public expenditures, but they do not include public investments, under which the building of new health centers is accounted. They are correlated with health expenditures, as they are also pro-cyclical -but they are not believed to respond to changes in child mortality, as they are not health-related expenditures, and follow political decisions based on other factors. Table 5 reports this additional exercise. Note: robust standard error in parentheses. ***, ** and * denote significance at 1%, 5% and 10% respectively. All variables in logarithm except "Political Dummy". The constant term is always included. No instruments included. Controls include all additional regressors present in Table 3 and Table 4.

Final Remarks
The literature often finds little or no impact of increased health expenditure on health status. Our results show that, when one focuses on mortality rates of children, improving the public health system does have an impact: mortality rates are lower when more health centers are available. This suggests that although overall mortality rates do not respond to changes in the public system, children can be effectively treated in it.
While we have focused so far on health centers, it is well-known that different types of physicians have different impacts on child mortality. Indeed, gynecologists seem to play a major role, which is related to the fact that prenatal care is the most effective in reducing hazards for newborns (Currie & Gruber, 1996).
Despite some preliminary results reflecting this impact, we have not been able to disentangle it effectively from the availability of health centers (which, as mentioned before, are an important workplace for various types of physician  (Hahn et al., 2011). We relate this exactly to the low variability across time of some regressors, such as income and education. Since fixed-effect estimators rely decisively on variation across time, it becomes imprecise. Alternatively, one could build longer series so as to obtain a large variability across time, which could lead to more precise fixed-effect estimates. This is one of the major challenges in this literature as many relevant data series change or are discontinued over time. Child mortality (DataSus): We use the ratio between child deaths before age five and the total population aged five years or less.

A1. Specification and Measurement of Variables
Income (