The use of simultaneous equation models in the empirical study of growth has so far been extremely limited. On the one hand, this is because appropriate admissible instruments are notoriously difficult to come by, as noted in the context of the GMM results presented above. On the other, there has been considerable reticence on the part of the profession to "go out on a limb" and specify an estimable structural form that stems from a well-specified theoretical model. In this section, two structural econometric models are considered in which a growth regression is paired with a "transmission mechanism" equation.
Although the Grossman model (1972) constitutes the foundation of the dynamic approach to the demand for health, as well as a theoretically sound basis for a model of life expectancy, relatively few growth theorists seem to be aware of its existence. In what follows, a continuous-time version of the "pure investment" Grossman model is sketched. This demonstrates how life expectancy may be endogenized within the context of a model of intertemporal optimization and, most importantly, provides solid theoretical foundations for the specification of a life expectancy equation which includes nutritional concerns among the explanatory factors.
In the "pure investment" Grossman (1972) model, the representative consumer allocates her revenue between a consumption good denoted here by c_{i}and investment in a stock of health capital, denoted by h_{t}.^{11} Investment in health capital obtains through expenditures, denoted here by p_{t}, which include medical expenditures per se as well other forms of expenditures which improve health. The consumer's budget constraint is thus given by:
(37)
where w represents the wage rate (assumed here to be constant) and l(.) is the individual's labour supply. The stock of health capital affects the individual's labour supply. The greater the stock of health capital, the smaller will be the amount of work time lost to ill health. This can be expressed by writing:
(38)
Grossman also assumes that there exists an upper bound on the labour supply of the individual, which can be expressed as:
(39)
A crucial assumption of the model is that there exists a lower bound on the stock of health capital of the individual compatible with survival. Let h represent this lower bound, which will be defined as follows:
(40) such that l(h)= 0.
Human physiology also plays a role in the evolution of the stock of health capital. Thus, the law of motion of health capital will include a rate of depreciation. It is likely that, ceteris paribus, this rate of depreciation will be quite small when an individual is young. However, it will tend to increase with age. This corresponds to the following expression:
(41)
The optimization problem faced by the representative agent is therefore given by:
(42) s.t.
(43) ,
(44) ,
where T represents the time of death of the individual, which will be endogenously determined within the model. Substituting from the budget constraint allows us to write the corresponding Hamiltonian as:
(45) ,
where represents the costate variable. By Pontryagin's Maximum Principle, the necessary conditions for an optimum are given by:
(46)
(47)
plus the usual transversality condition. The first condition implies that the costate variable can be expressed as:
(48) .
Differentiating with respect to time and rearranging yields:
(49) .
The laws of motion of the system are therefore given by:
(50) ,
and
(51) .
The steady-state conditions then imply that:
(52) ,
and
(53) ,
where, in contrast to what one usually obtains in a growth model, the steady-state stock of health capital is a function of time (expressed by writing h^{*}_{t} instead of h^{*} ) because of the dependence of the depreciation rate of health capital on t. From the first condition, it follows that:
(54) .
Sincel_{h} (h_{t}) is a decreasing function, it follows that its inverse, l^{-1}_{h} (.) , is so as well. If one considers an ad hoc specification in which the depreciation rate of health capital is a function not only of time but of some measure of the extent of malnutrition, such as the DES per caput, that is , with , it follows that the steady-state level of health capital, and thus life expectancy, will be an increasing function of the DES per caput. This provides a simple theoretical justification for the inclusion of nutritional variables in a life expectancy equation.
An additional justification for including the DES per caput in a life expectancy equation is provided by several well-known micro-econometric studies that have examined the question, although it should be noted that the focal point of this work has been concentrated on infant mortality. Martorell and Ho (1984), in their survey of the literature, note that "the studies reviewed here all show that severely malnourished children...have greatly increased mortality risks relative to normal children. Children with mild and moderate malnutrition...also showed increased mortality risks."^{12}(p. 61) Since the variable used in the cross-country growth regressions is life expectancy at age zero, the increased mortality risk of malnourished children should be captured by this variable. The interested reader is referred to the excellent survey by Behrman and Deolalikar (1988) for more details on the relevant literature.
The Grossman model, as well as the micro-econometric evidence, suggest that nutritional concerns should be an important determinant of life expectancy at the aggregate, cross-country level. This intuition is confirmed empirically by an extremely simple regression presented in column (1) of Table 11, in which life expectancy at age zero is regressed on the two decade dummies, the two continent dummies, the initial level of GDP per caput, the DES per caput, the number of physicians per 10,000 inhabitants, and the growth rate of GDP per caput. The coefficient associated with the DES per caput is highly significant at the usual levels of confidence.^{13} Moreover, there is a potentially interesting "feedback" effect of the growth rate of GDP per caput on life expectancy. Essentially, in countries that growth faster, individuals live longer : an increase of the growth rate of GDP per caput of one percentage point (0.01) increases life expectancy at age zero by approximately six months. Both of these effects obtain, it should be noted, even though GDP per caput is included in the specification. This regression also performs remarkably well in explaining differences in life expectancy across countries : the associated adjusted R^{2 } is equal to 0.852.
TABLE 11
PFI, DES per caput, and economic growth
Transmission mechanisms : life expectancy
Estimation of a two-equation structural model
(t-statistics below coefficients)
(1) |
(2) |
(3) |
|||||
Dependent variable |
Life expectancy at age zero |
Growth rate of GDP per caput |
Life expectancy at age zero |
Growth rate of GDP per caput |
Life expectancy at age zero |
||
Method of estimation |
OLS |
GMM |
GMM |
||||
Intercept |
9.538 |
-0.112 |
9.788 |
-0.111 |
9.613 |
||
2.171 |
-3.303 |
2.183 |
-3.318 |
2.194 |
|||
1960s dummy |
-3.113 |
0.024 |
-4.653 |
0.024 |
-4.664 |
||
-3.559 |
5.430 |
-3.742 |
6.997 |
-3.758 |
|||
1970s dummy |
-1.479 |
0.017 |
-2.710 |
0.018 |
-2.719 |
||
-1.717 |
4.405 |
-2.499 |
4.860 |
-2.513 |
|||
Africa dummy |
-5.976 |
-0.002 |
-5.079 |
-5.154 |
|||
-6.479 |
-0.196 |
-4.220 |
-4.526 |
||||
Latin America dummy |
1.372 |
-0.014 |
2.261 |
-0.014 |
2.238 |
||
1.514 |
-4.159 |
2.048 |
-4.113 |
2.043 |
|||
Log of initial GDP |
5.231 |
-0.011 |
5.344 |
-0.012 |
5.370 |
||
7.859 |
-1.567 |
7.353 |
-2.443 |
7.526 |
|||
Life expectancy at age zero |
0.002 |
0.002 |
|||||
1.523 |
3.170 |
||||||
DES per caput (kcal/day) |
4.50E-03 |
8.45E-05 |
0.004 |
8.19E-05 |
0.004 |
||
3.786 |
2.987 |
3.467 |
3.271 |
3.532 |
|||
DES per caput, squared |
-1.53E-08 |
-1.50E-08 |
|||||
-3.392 |
-3.553 |
||||||
Physicians per 10,000 inhab. |
0.154 |
0.153 |
0.146 |
||||
2.289 |
2.185 |
2.525 |
|||||
Growth rate of GDP per caput |
54.62 |
129.03 |
128.64 |
||||
3.722 |
2.733 |
2.729 |
|||||
Mean of dependent variable |
58.40 |
0.022 |
58.40 |
0.022 |
58.40 |
||
Test of overidentifying restrictions: (degrees of freedom) p-value |
n.a. |
0.846 |
|||||
Number of observations |
215 |
215 |
215 |
Note: in column (1), adjusted R^{2} = 0.852, = 4.373; method of estimation for systems (2) and (3), Generalized Method of Moments using exclusion restrictions in order to identify. Data source. Physicians per 10 000 inhabitants : World Bank Social Indicators.
In the next two columns in Table 11, results are presented corresponding to GMM estimation of the two-equation system constituted by a growth regression that mirrors that presented in column (7) of Table 7 (pooling results including life expectancy, the difference here being that life expectancy is assumed to be endogenous) as well as a life expectancy equation similar to the one presented in column (1) of Table 11. In the growth equation, it is striking that all coefficients are similar in size to those presented in column (7) of Table 7 (despite the loss of 76 observations because of the inclusion of our exogenous instrument, physicians per 10 000 inhabitants), with the possible exception of the coefficient associated with the SSA dummy which is now statistically indistinguishable from zero. In particular the coefficients associated with the DES per caput and the DES per caput squared are both statistically indistinguishable from those presented in the simple pooling results. In the life expectancy equation, on the other hand, the feedback effect of the growth rate of GDP per caput on life expectancy is now substantially larger : unsurprisingly, the OLS results presented in column (1) of Table 11, in which the growth rate of GDP per caput was assumed to be exogenous, tended to underestimate this effect. On the other hand, the impact of the DES per caput on life expectancy is remains statistically significant and of the same order of magnitude.
The identifying restriction that allows one to estimate this system by GMM is, of course, the inclusion of physicians per 10,000 inhabitants in the life expectancy equation, and its exclusion from the growth regression. Note that, when introduced into a growth regression, the coefficient associated with this variable is statistically indistinguishable from zero, lending some support to its use as an exogenous instrument. With the system presented in column (2), it is, of course, impossible to formally test the validity of this instrument given that the system is just identified.
In column (3) of the same Table, results are presented corresponding to the most parsimonious specification possible in the sense that all insignificant variables whose exclusion does not change the coefficients of interest in a statistically significant manner are dropped. In this case, the system is overidentified with 4 degrees of freedom and the standard Sargan test can be used in order to assess the validity of our set of overidentifying restrictions. The p-value associated with this test is equal to 0.846, implying that the appropriateness of the instruments cannot be rejected. This lends substantial statistical support to our results.
The results presented in column (3) of Table 11 allow one to carry out a first diagnostic regarding the relative magnitude of the indirect versus direct effects of the DES per caput on the growth rate of GDP per caput. Write the growth regression as:
(55) ,
where Y_{it} is the growth rate of GDP per caput, Z_{it } is the vector of controls, X_{it} is the DES per caput, and L_{it} is life expectancy at age zero. Similarly, write the life expectancy equation as:
(56) ,
where A_{it} is the vector of control variables in the life expectancy equation. Then the direct impact of the DES per caput on growth is given by:
(57) ,
whereas its indirect impact, operating through life expectancy, is equal to:
(58) .
If the feedback effect s are included, the effects are given, respectively, by:
(59) and .
TABLE 12
PFI, DES per caput, and economic growth
Transmission mechanisms : life expectancy and schooling
Estimation of a three-equation structural model
Method of estimation : Generalized Method of Moments (GMM)
(t-statistics below coefficients)
(1) |
(2) |
(3) |
|||||||||
Dependent variable |
Growth rate of |
Life ex-pectancy |
Log of years schooling |
Growth rate of |
Life ex-pectancy |
Log of years schooling |
Growth rate of |
Life ex-pectancy |
Log of years schooling |
||
Exogeneity assumption |
Log schooling |
Log schooling |
Log schooling |
||||||||
Intercept |
-0.179 |
4.142 |
1.940 |
0.001 |
-4.293 |
5.873 |
-0.246 |
||||
-3.128 |
0.984 |
1.099 |
0.051 |
-1.003 |
2.631 |
-9.743 |
|||||
1960s dummy |
0.025 |
-3.498 |
-0.002 |
0.024 |
-6.074 |
-0.561 |
0.024 |
-2.360 |
|||
5.350 |
-3.222 |
-0.017 |
5.291 |
-5.325 |
-3.640 |
6.564 |
-3.830 |
||||
1970s dummy |
0.013 |
-1.738 |
-0.077 |
0.018 |
-4.486 |
-0.553 |
0.012 |
||||
2.927 |
-1.755 |
-0.759 |
4.189 |
-3.795 |
-3.655 |
3.802 |
|||||
Africa dummy |
0.000 |
-3.887 |
-0.148 |
0.004 |
-1.587 |
0.135 |
-4.040 |
-0.127 |
|||
-0.011 |
-3.208 |
-1.511 |
0.556 |
-1.043 |
0.883 |
-5.372 |
-2.430 |
||||
Latin America dummy |
-0.013 |
1.759 |
0.037 |
-0.016 |
4.134 |
0.412 |
-0.009 |
||||
-3.201 |
1.742 |
0.382 |
-3.844 |
3.448 |
3.476 |
-3.875 |
|||||
Log of initial GDP |
-0.007 |
6.316 |
-0.643 |
-0.028 |
7.627 |
-1.614 |
7.113 |
-0.134 |
|||
-0.624 |
9.815 |
-1.395 |
-6.802 |
10.467 |
-2.888 |
28.657 |
-4.984 |
||||
Log of initial GDP, squared |
0.072 |
0.142 |
0.041 |
||||||||
2.388 |
3.829 |
12.716 |
|||||||||
Life expectancy at age zero |
0.003 |
0.003 |
0.004 |
||||||||
2.317 |
3.662 |
6.016 |
|||||||||
Log schooling |
-0.038 |
0.004 |
-0.066 |
||||||||
-2.180 |
0.731 |
-7.235 |
|||||||||
DES per caput (kcal/day) |
7.8E-05 |
2.9E-03 |
5.8E-05 |
1.7E-05 |
0.002 |
-3.4E-04 |
8.1E-05 |
2.2E-03 |
|||
2.008 |
2.995 |
0.611 |
1.263 |
1.260 |
-1.896 |
3.243 |
2.863 |
||||
DES per caput, squared |
-1.4E-08 |
-4.1E-09 |
-1.5E-08 |
||||||||
-2.331 |
-2.518 |
-3.777 |
|||||||||
Physicians per 10,000 inhab. |
0.115 |
0.030 |
0.082 |
||||||||
2.156 |
1.068 |
2.989 |
|||||||||
Growth rate of GDP per caput |
115.98 |
-0.840 |
254.57 |
26.026 |
87.059 |
||||||
3.275 |
-0.191 |
8.455 |
4.686 |
4.007 |
|||||||
Mean of dependent variable |
0.023 |
60.583 |
1.337 |
0.023 |
60.583 |
1.337 |
0.023 |
60.583 |
1.337 |
||
Test of over-identifying restriction: (degrees of freedom) p-value |
(df = 2) 0.381 |
(df = 5) 0.000 |
(df =13) 0.637 |
||||||||
Number of observations |
178 |
178 |
178 |
Note : test of overidentifying restrictions in the specification corresponding to the parsimonious system (3), but where log schooling is assumed to be exogenous has a p-value of 0.000.
Using equations (57) and (58), it can be seen that raising the DES per caput in all countries to 2770 kcal /day (the average level, suggested in the Sixth World Food Survey, at which the PFI would fall to zero) in those countries where the DES per caput is below this level would increase their annual growth rate of GDP per caput by 0.43 percentage points directly and by an additional 0.33 percentage points through the impact mediated by life expectancy. When the feedback effect of growth on life expectancy is included in the calculations (equation (59)), the corresponding figures are 0.55% and 0.42%. The total impact on the annual growth rate of per caput GDP is thus comprised between 0.77 and 0.97 percentage points, divided approximately equally between direct and indirect effects.
Table 12 presents empirical results corresponding to a three-equation system constituted by (i) a life expectancy equation, (ii) a schooling equation, and (iii) a growth equation. Both life expectancy and schooling are assumed to be endogenous in the results presented in columns (1) and (3), whereas those results presented in column (2) correspond to a specification in which schooling is assumed to be exogenous.
The results, particularly with respect to the usual assumption of exogeneity of schooling in growth regressions, are particularly interesting. When schooling is assumed to be exogenous (column (2) in Table 12), three results obtain. First, the coefficient associated with schooling is statistically indistinguishable from zero in the growth equation. Second, convergence effects are of the usual negative sign and highly significant. Third, the coefficient associated with life expectancy in the growth equation is positive and of the same order of magnitude as for those results presented in Table 11.
These results, while interesting, are not robust empirically. This is because the overidentifying restrictions (with 5 degrees of freedom) are soundly rejected with a corresponding p-value that is smaller than 0.001. In contrast, when the same system is estimated assuming that schooling is endogenous in the growth regression, the test of the overidentifying restrictions is not rejected (p-value = 0.381). Moreover, convergence effects vanish in the growth regression and the coefficient associated with schooling in the same equation is now negative and statistically significant. Column (3) of the same Table presents results corresponding to the most parsimonious specification of the three equation system, in which schooling is assumed to be endogenous. Those results presented in column (1) are, unsurprisingly, reinforced by the increase in degrees of freedom and, as with the full system in which schooling is endogenous, the specification is not rejected by a test of the overidentifying restrictions (p-value = 0.637). In terms of the DES per caput variables in the growth regression, the quadratic specification is estimated quite precisely, as is the impact of the DES per caput in the life expectancy equation.
A possible source of concern in the results presented in Table 12 is that the schooling equation is mis-specified. Given the manner in which the GMM procedure exploits the correlation among equations, it is therefore quite possible that specification errors in the schooling equation are transmitted via the variance-covariance matrix to the coefficients estimated in the life expectancy and growth equations.
In order to assess whether this was indeed the case, Table 13 presents results using the two-equation specification of Table 11, but in which schooling is now included as an additional explanatory variable in the growth regression. Two sets of results are presented, corresponding to the schooling variable being either endogenous (columns (1) and (3)) or exogenous (columns (2) and (4)) in the growth equation.
Broadly speaking, the results corroborate those presented in Tables 11 and 12. First, with schooling assumed endogenous, the convergence effects in the growth regression vanish. Second, the quadratic specification in terms of the DES per caput in the growth regression remains statistically significant, particularly so in the parsimonious specification presented in column (3). Third, the coefficient associated with schooling is negative and statistically significant in the system corresponding to column (1) and remains so in the parsimonious specification presented in column (3). Fifth, the specifications in which schooling is assumed to be exogenous are strongly rejected by the test of the overidentifying restrictions, while the opposite is true when schooling is assumed endogenous (p-values equal to 0.184 and 0.412, respectively, in columns (1) and (3)).
TABLE 13
PFI, DES per caput, and economic growth
Transmission mechanisms : life expectancy
Estimation of a two-equation structural model
Method of estimation : Generalized Method of Moments (GMM)
(t-statistics below coefficients)
(1) |
(2) |
(3) |
(4) |
||||||||
Dependent variable |
Growth rate of |
Life expectancy |
Growth rate of |
Life expectancy |
Growth rate of |
Life expectancy |
Growth rate of |
Life expectancy |
|||
Exogeneity assumption |
Log schooling |
Log schooling |
Log schooling |
Log schooling |
|||||||
Intercept |
-0.175 |
4.680 |
0.016 |
-7.103 |
-0.177 |
4.724 |
0.009 |
-9.399 |
|||
-2.988 |
1.063 |
0.723 |
-1.750 |
-3.009 |
1.077 |
0.550 |
-2.229 |
||||
1960s dummy |
0.025 |
-3.434 |
0.025 |
-5.329 |
0.025 |
-3.423 |
0.023 |
-5.729 |
|||
5.267 |
-3.153 |
4.908 |
-4.770 |
6.479 |
-3.160 |
5.705 |
-4.811 |
||||
1970s dummy |
0.013 |
-1.718 |
0.018 |
-4.025 |
0.013 |
-1.709 |
0.018 |
-4.488 |
|||
2.954 |
-1.739 |
3.758 |
-3.570 |
2.972 |
-1.740 |
4.063 |
-3.657 |
||||
Africa dummy |
0.001 |
-3.862 |
0.010 |
-1.938 |
-3.845 |
-0.002 |
|||||
0.076 |
-3.191 |
1.180 |
-1.344 |
-3.234 |
-0.002 |
||||||
Latin America dummy |
-0.013 |
1.754 |
-0.014 |
3.192 |
-0.013 |
1.749 |
-0.016 |
4.043 |
|||
-3.178 |
1.739 |
3.025 |
2.766 |
-3.166 |
1.734 |
-3.732 |
3.559 |
||||
Log of initial GDP |
-0.007 |
6.235 |
-0.034 |
7.819 |
-0.006 |
6.231 |
-0.029 |
8.165 |
|||
-0.694 |
9.281 |
-6.692 |
11.418 |
-0.642 |
9.286 |
-6.983 |
11.223 |
||||
Life expectancy at age zero |
0.003 |
0.005 |
0.003 |
0.004 |
|||||||
2.245 |
4.246 |
3.811 |
5.029 |
||||||||
Log schooling |
-0.038 |
-0.004 |
-0.040 |
-0.002 |
|||||||
-2.165 |
-0.581 |
-2.251 |
-0.482 |
||||||||
DES per caput (kcal/day) |
7.3E-05 |
2.9E-03 |
-1.1E-05 |
2.5E-03 |
7.2E-05 |
2.9E-03 |
7.9E-06 |
1.9E-03 |
|||
1.635 |
2.988 |
-0.809 |
2.231 |
1.957 |
2.994 |
0.615 |
1.519 |
||||
DES per caput, squared |
-1.3E-08 |
-1.7E-10 |
-1.3E-08 |
-2.6E-09 |
|||||||
-1.943 |
-0.105 |
-2.186 |
-1.561 |
||||||||
Physicians per 10 000 inhab. |
0.131 |
-0.013 |
0.133 |
0.001 |
|||||||
1.965 |
-0.526 |
2.013 |
0.060 |
||||||||
Growth rate of GDP per caput |
113.81 |
222.24 |
113.66 |
258.62 |
|||||||
3.203 |
7.197 |
3.201 |
10.415 |
||||||||
Mean of dependent variable |
0.023 |
60.583 |
0.023 |
60.583 |
0.023 |
60.583 |
0.023 |
60.583 |
|||
Test of overidentifying restriction: (degrees of freedom) p-value |
(df=1) 0.184 |
(df=3) 0.000 |
(df=2) 0.412 |
(df=4) 0.000 |
|||||||
Number of observations |
178 |
178 |
178 |
178 |
These results confirm our previous findings of the existence of both a direct impact of the DES per caput on the growth rate of GDP per caput, as well as an indirect effect that operates through the impact of nutrition on life expectancy. They also raise serious doubts concerning the concept of conditional convergence since, once schooling is assumed endogenous in the growth equation, the coefficient associated with initial GDP per caput becomes statistically indistinguishable from zero. Finally, they imply that convergence effects would appear to operate through differences in schooling, since the coefficient associated with this variable is negative and statistically significant at the usual levels of confidence.
In quantitative terms, the results presented in system (3) confirm the validity of the quadratic specification in terms of the DES per caput (although it should be noted that this specification is sensitive to the inclusion of country-specific fixed effects or first differencing). The critical value of the DES per caput at which the impact of nutrition on growth becomes negative in this case is equal to 2732 kcal /day, which is remarkably close to the level -2770 kcal / day- estimated by the Sixth World Food Survey as being sufficient to eliminate food inadequacy altogether.
11 In his original formulation Grossman consider the time allocation problem faced by the consumer as well, and this within a discrete-time framework; what follows is a simplified, continuous-time treatment of the model.
12 See Pelletier et al (1994) for a recent contribution. It should be noted that Horton et al's (1985) cross-country empirical results on the determinants of infant mortality are not particularly convincing when it comes to the impact of the price of "cheap calories". As with many of these cross-country studies done in the 1980s, the problem may simply lie with the poor quality of the data that were available at the time.
13 Note that when Preston (1980), in a well-known paper, included the excess of the DES above 1500 kcal /day in a life expectancy equation, he found the associated coefficient to be insignificant at the usual levels of confidence. This result remains unchanged when Preston moved to country-specific fixed effects.