The simple theoretical constructs presented in section 1 in order to account for the statistical significance of nutritional variables in the simple growth regressions suffer, of course, from the assumption of exogeneity, in the sense that the impact of malnutrition on growth is not modelled explicitly within the theoretical model. In what follows, a simple endogenous growth model is constructed in which the productivity of labour depends on consumption, and in which nutritional growth traps can endogenously arise. This theoretical model is then used as the basis for an empirical procedure in which the response of growth to nutritional concerns is conditional on the nutritional status of the population.
Consider the simple AK model sketched in section 1 above. Let effective labour input be specified as:
(21)
This formulation implies that effective labour input is an increasing function of consumption for , whereas, once a minimal level of consumption is attained (denoted here by ), the usual dynamics of the AK model take over. It is worth noting that this formulation is more faithful to the notion of nutritional efficiency wage effects than the simple ad hoc formulation presented in section 1, since the productivity of labour, via the efficiency impact of consumption, is endogenously determined within the model.
With this parameterization, the dynamics of capital are given by:
The corresponding Hamiltonian is therefore given by:
(22)
By an application of Pontryagin's Maximum Principle, the necessary conditions for an optimum are given by:
(23)
(24)
plus the usual transversality condition. From the first condition, the solution for the costate variable is:
(25)
Taking its derivative with respect to time, the following expression results:
After some manipulations, this can be rewritten as:
(26)
Now from the second condition, another expression can be obtained for the growth rate of the costate variable. Equating the two expressions allows the following expression:
The laws of motion of the system are therefore given by:
(27)
(28)
Notice that, for = 0, the usual expression of a constant (positive) growth rate of consumption (see equation (27)) can be obtained. The question is, of course, whether the dynamics of the system will allow the economy to reach the level of consumption at which at which point the usual growth dynamics of the AK model take over.
Consider the equation that governs the dynamics of consumption. Then it is clear that to reach a situation where , it must be the case that:
For both the numerator and the denominator of this expression to be negative (the other possibility being that they are both positive), it must be the case that:
(29)
The question then is whether such an interval exists, which boils down to whether the inequality:
(30)
can hold. For very low initial levels of consumption, which corresponds empirically to a large shortfall of c_{t} with respect to , one can write, evaluating the preceding expression at , , which of course cannot hold. It follows that it is quite possible, for low initial levels of consumption, which correspond to a very high PFI or a very low DES per caput, that the growth rate of consumption will be negative and that the economy will remain in an "undernourishment growth trap".
What are the empirical implications of the presence of nutritional growth traps? A priori, the most probable manner in which one can identify a country subject to a nutritional growth trap is in terms of the absence of convergence effects. While a standard growth model yields a convergence equation in which the subsequent rate of growth is a decreasing function of the initial level of GDP per caput (see equation (12)), one would expect, for a country caught in a growth trap, that such an effect would not obtain, meaning that the coefficient associated with initial GDP per caput should be statistically indistinguishable from zero in a growth regression. In addition, it is for countries caught in a nutritional growth trap that the impact of the DES per caput is most likely to be felt. Moreover, it is clear from the theoretical model presented above, that for a country with a level of consumption outside of the range in which the growth trap effect operates (in terms of the model, this means that in such countries ), that there should be no impact of the DES per caput on the productivity of labour. The DES per caput should therefore be excluded from the growth regression corresponding to such countries. These exclusion restrictions will constitute the basis for the empirical implementation pursued in what follows.
In order to test for the presence of nutritional growth traps in a manner that corresponds to the theoretical arguments enunciated above, we turn to a switching regression specification in which regime 1 corresponds to a "high PFI" scenario and regime 2 corresponds to a "low PFI" scenario. Whether a country belongs to regime 1 or regime 2 will depend simultaneously upon its income per caput and its PFI.
Let x_{i} (i=1, ....N, indexes observations) denote the N X 5 matrix of control variables that is common to both regimes (a constant term, two decade dummies, and two continent dummies). Let x_{1i}=[x_{i} DES_{i}] denote the N × 6 matrix of explanatory variables that correspond to the high PFI regime and x_{2i}=[x_{i} log GDP/capita_{i} log Schooling] denote the N × 7 matrix of explanatory variables that corresponds to the low PFI regime; the dependent variable (the annual growth rate of per caput GDP) will be denoted by y_{i} with _{ }y_{1i} denoting the growth rate under the high PFI regime and y_{2i} denoting the growth rate under low PFI regime. The econometric specification is then given by the following system of equations:
(31)
where the choice between regimes is given by the following sorting condition:
(32)
with , and where z_{i} is an N x 3 matrix constituted by a constant term, the logarithm of GDP per caput, and the logarithm of the PFI. The distributional assumptions on the disturbance terms in the two regimes are given by where:
(33)
It is well known (e.g., Maddala (1983), Quandt (1988)) in the context of this type of model that the offdiagonal term _{12} is not identifiable (for the simple reason that both regimes cannot simultaneously obtain). Note that, for identification purposes (Maddala and Nelson (1975), p. 424), it is necessary to (i) normalize the standard deviation of the selection equation such that it is equal to one (that is why n_{i} is distributed N(0,1) and not and (ii) normalize the coefficient on the constant term in the selection equation to one (i.e., the parameters in the selection equation are only identified up to a multiplicative constant).
The selection equation which determines whether a country belongs to the high PFI regime or the low PFI regime is given by a (latent) regime indicator function defined as follows:
(34)
Therefore the lefthandside variable from the growth equations can be written in the following form:
(35)
where I_{i}(z_{i}) is approximated in continuous form by the probit function:
(36)
One can then easily construct the corresponding likelihood function and maximize it with respect to _{i}, _{2}, _{1}, _{2} , which, owing to the above normalizations, are all identifiable. Note that this procedure does not arbitrarily assign an observation to a given regime: this process is carried out optimally through the maximization of the likelihood function, thus allowing the data (conditioned by the variables included in the selection equation) to sort themselves freely into the two regimes.
Results are presented in Table 9. As is obvious from the estimated coefficients, sample separation based on the logarithm of initial GDP per caput and the PFI is crisp, with the associated coefficients estimated quite precisely. The sample separation based on initial GDP per caput and the PFI is illustrated graphically in Figure 6.
In the low PFI regime, convergence effects are strong, and the logarithm of schooling appears with the usual positive, and statistically significant coefficient. In the high PFI regime, on the other hand, the coefficient associated with the DES per caput is equal to 2.56 x 10^{5}, of similar order of magnitude to that obtained in column (4) of Table 8 which corresponds to the simple pooling results that include schooling. Here, the coefficient is twice as large, which is to be expected given that it is estimated only on those observations which the maximum likelihood procedure classified as belonging to the high PFI regime.
TABLE 9
Nutritional growth traps ?
Estimation of a tworegime (high PFI, low PFI) switching regression model
with unknown sample separation
(tstatistics below coefficients)
(1) 
(2) 

Selection equation 

Log of initial GDP (by decade) 
0.300 
0.297 

3.924 
3.652 

PFI (%) 
0.268 
0.266 

4.618 
4.214 

Growth equation 
Low PFI regime 
High PFI regime 
Low PFI regime 
High PFI regime 

Intercept 
0.184 
0.053 
0.183 
0.045 

5.771 
2.718 
5.629 
1.598 

1960s dummy 
0.014 
0.025 
0.014 
0.025 

2.891 
5.663 
2.799 
4.904 

1970s dummy 
0.019 
0.017 
0.018 
0.017  
4.697 
4.186 
4.591 
3.639 

Africa dummy 
0.033 
0.010 
0.032 
0.011 

1.501 
2.203 
1.524 
2.166 

Latin America dummy 
0.019 
0.013 
0.019 
0.012 

4.775 
2.340 
4.678 
1.983 

Log of initial GDP (by decade) 
0.023 
0.023 
0.001 

5.083 
4.938 
0.325 

Log schooling 
0.019 
0.018 

3.016 
2.865 

DES per caput (kcal/day) 
2.56E05 
2.59E05 

3.208 
2.852 

Mean of dependent variable 
0.0300 
0.0119 
0.0300 
0.0119 

Average estimated probability 
0.459 
0.540 
0.458 
0.541 

0.017 
0.015 
0.017 
0.015 

14.294 
12.053 
14.273 
12.153 

Log likelihood 
650.458 
650.511 

Number of observations 
251 
251 
Note: estimated by Maximum Likelihood. Data source : same as Table 8.
Perhaps the most comforting result is that the mean value of the growth rate of GDP per caput for those countries classified as belonging to the low PFI regime (0.030) is almost three times as large as the mean value of the growth rate of those countries classified as belonging to the high PFI regime (0.012), as one would expect if the underlying theoretical model is an adequate representation of reality. Finally, note that the theoretically motivated assumption that there are no convergence effects in the regime associated with a nutritional growth trap is supported by the results presented in the second part of Table 9 (system (2)) where the logarithm of initial GDP per caput in the growth equation corresponding to the high PFI regime is included. The coefficient associated with log initial GDP per caput is equal to 0.001 with a corresponding tstatistic of 0.325. Thus, one of our identifying assumptions, namely the absence of convergence effects in the high PFI regime, is supported statistically by our empirical results.
Figure 6
Estimation of a tworegime (high PFI, low PFI) switching regression
Sample separation as determined by log GDP per caput and the PFI
Note : points to the southeast of the line are countrydecades classified by the switching regression presented in Table 9 (system (1)) as belonging to the high PFI regime; points to the northwest are countrydecades classified as belonging to the low PFI regime.
TABLE 10
Nutritional growth traps?
Classification of countries by switching regression procedure
Countries belonging to a given regime during all decades for which data are available 

Countries stuck in high PFI regime through all decades for which data are available 
Afghanistan, Angola, Burundi, Benin, Burkina Faso, Bangladesh, Bolivia, Botswana, Central African Republic, Cameroon, Congo, Dominican Republic, Ecuador, Ethiopia, Gabon, Ghana, The Gambia, Guatemala, Honduras, Haiti, India, Jamaica, Kenya, Liberia, Sri Lanka, Lesotho, Madagascar, Mali, Mozambique, Mauritania, Mauritius, Malawi, Niger, Nigeria, Nicaragua, Nepal, Pakistan, Peru, Philippines, Rwanda, Sudan, Senegal, Sierra Leone, El Salvador, Somalia, Suriname, Chad, Togo, Thailand, Tanzania, Uganda, Republic of Yemen, Zaire, Zambia, Zimbabwe 
Countries in low PFI regime through all decades for which data are available 
Argentina, Australia, Austria, Belgium, Brazil, Barbados, Canada, Switzerland, Cyprus, Germany, Denmark, Spain, Finland, France, United Kingdom, Greece, Hong Kong, Ireland, Iceland, Israel, Italy, Jordan, Japan, Republic of Korea, Luxembourg, Mexico, Malta, Malaysia, Netherlands, Norway, New Zealand, Portugal, Paraguay, Singapore, Sweden, Trinidad and Tobago, Turkey, Uruguay, United States 
Countries changing regime between decades 

Emerged from high PFI regime in the 70s 
Algeria, Egypt, Morocco, Papua New Guinea, Syrian Arab Republic, Tunisia 
Emerged from high PFI regime in the 80s 
Myanmar, Colombia, Costa Rica, Indonesia, Swaziland 
Emerged from high PFI regime in 70s, but returned to high PFI regime in 80s 
Côte d'Ivoire, Guyana, Venezuela 
Fell into high PFI regime in 80s 
Chile, Panama 
In summary, the switching regression results suggest that nutritional growth traps exist, and that the quadratic specification uncovered in the simple pooling results stems from the coexistence of two regimes, one in which the PFI is high and in which the impact of the DES per caput on growth is large, a second in which the PFI is low and in which the impact of the DES per caput on growth is absent. The switching regression results also imply that a gap of 1.8 percentage points in the annual growth rate of GDP per caput can be accounted for by a country belonging to the high PFI regime.
Although it is wise to be cautious regarding econometric results based on a latent variables approach, it is interesting to examine how various observations (countrydecades) are classified by our switching regression procedure. Apart from the set of countries that were members of the OECD in the 1960s, which are, unsurprisingly, classified into the low PFI regime, a certain number of countries stand out in the low PFI subsample. Details are provided in Table 10. For example Brazil, Jordan, the Republic of Korea, Mexico, Malaysia, Paraguay, Trinidad and Tobago and Uruguay are all classified into the low PFI regime starting with the 1960s. Botswana, Thailand and the Philippines, on the other hand, are classified in the high PFI regime into the 80s despite impressive growth rates of GDP, suggesting that inequality and the ensuing high rates of malnutrition have constrained what might have otherwise been even higher rates of growth.
The most interesting cases are provided by those countries that moved from the high PFI regime to the low PFI regime, indicating that those gains to growth attainable by improving the nutritional status of the population were exhausted. A group of five Middle East and North Africa (MENA) countries (Algeria, Egypt, Morocco, Syrian Arab Republic and Tunisia), plus Papua New Guinea emerged from the high PFI regime in the 70s, an indication that the constraints imposed on growth by inadequate nutrition (but not necessarily other constraints) were probably eliminated by the massive social programmes implemented in those Arab countries following the first oil price shock. This first group of countries is joined in the low PFI regime in the 80s by Myanmar, Colombia, Costa Rica, Indonesia and Swaziland. Côte d'Ivoire, Guyana, Venezuela are classified as having emerged from the high PFI regime during the 1970s only to return to it during the 1980s. Finally, Panama and Chile, after being classified as belonging to the low PFI regime for the 1960s and 1970s, are classified as having fallen into the high PFI regime during the 1980s.
It is worth stressing that being classified as belonging to the high PFI regime does not condemn a country to low rates of growth. What it does mean is that nutritional factors are important for such countries and that there is room for improving the growth rate of GDP per caput by increasing the DES per caput. In countries operating within the low PFI regime, on the other hand, nutritional concerns are of much more limited concern, and there remains no room to improve growth performance by increasing the DES per caput.