Lastly, having found evidence of the existence of the shortrun mechanisms suggested by clas-sical growth theory, an additional coirf will be brie y interpreted, as it is regularly used in the prevailing empirical literature. The eect that might be important to consider arises from the statistically signi cant negative lagged response of death rate to changes in GDP per capita growth as is illustrated by the respective central right coirfs of Figures 7.3 and 7.4. So far, exogeneity of the death rate has been assumed to trigger the epidemiological transition. As
a shortterm relation, however, GDP per capita growth seems to have aected mortality even before the epidemiological transition, since this eect can be found to prevail in the early English data sample (see Figure 7.5, upper right graph) and to wear o in the late sample (see Figure 7.5, lower right graph).
Some authors (Nicolini 2007; Crafts and Mills 2009; P ster and Fertig 2010; Fernihough 2012;
Herzer et al. 2012; Moller and Sharp 2014; Rathke and Sarferaz 2014; Edvinsson 2017) have argued in favor of complementing Malthusian eects in the sense that higher productivity not only enhances fertility, but at the same time operates towards lower mortality. These authors regard the statistically signi cant positive eect of GDP per capita growth on birth rate corresponding to the eect of the PoP in this work as evidence of preventive checks in general, which are not to be confused with the PoG. Accordingly, if income was observed to raise births on average, it is a sign that reproduction has formerly been suppressed by preventive fertility behavior. Equally, they hold the apparent negative causal relationship between GDP per capita and death rate to universally re ect positive checks. Their idea is that whenever living standards would fall below a subsistence level, the positive checks are supposed to increase the death rate as a general result of individuals heavily competing for the remaining resources.
This eect deserves attention and could be added to the simulation to complement the mechanism of stagnation by providing another channel of population growth. In this paper, the modeling of the eect of conventional positive checks has been disregarded for two reasons. Firstly, it does not provide explanatory power for the mechanism of growth, since the positive checks are thought to disappear at the same time as GDP per capita rose above subsistence level. When modeling and evaluating the growth regime, it is regarded to be su cient to focus on the steady decline of fertility as the crucial factor contributing to the population slowdown inducing the breakout from stagnation. Secondly and more importantly, the current conventional cliometric interpretation of preventive and positive checks is at odds with Malthus' de nition that
the preventive check is perhaps best measured by the smallness of the proportion of yearly births to the whole population,9
i.e. by the level of the birth rate and that
the positive checks to population [. . . ] include every cause [. . . ] which in any degree con-tributes to shorten the natural duration of life,10
which are best measured by the level of the death rate. Consequently, the preventive checks ought not to be measured by the causal relationship running from GDP per capita to fertility, which is reserved for the PoP. Instead, it might be very generally concluded that a low birth rate is a sign of the operation of preventive checks, whereas a high death rate reveals the operation of positive checks. Naturally, this implies an important Malthusian insight that has already been hinted at that the regime of stagnation is characterized by high mortality and the regime of development by low fertility.
9 Malthus (1826), book II, chapter XI.
10 Malthus (1826), book I, chapter II.
7.5 Concluding Remarks on the Empirics of the Theory of Pop-ulation
Since it was considered insu cient to construct a model that merely ts the stylized historical facts of stagnation and growth, the operation of the classical principles has in this chapter been evaluated collectively to avoid the reasonable impression of reverse engineering in our simula-tions. To this end, a simple VAR estimation provided a way to establish evidence of the suggested classical shortrun relationships by employing orthogonalized cumulative impulse response func-tions derived from two historical samples, based on approximately 4,500 observafunc-tions of annual national data on birth rate, death rate and GDP per capita growth. In those cases, in which the principles were a priori supposed to be measurable, in particular for the PoG and the PoP, the impulse responses yielded strong support for the theory. Additional robustness tests conducted with regard to countryspeci c eects and timevarying coe cients were generally in line with the classical principles. Also, it has been suggested that recent publications might require recon-sideration regarding the use of positive checks and preventive checks, as they seem to be at odds with Malthus' original terminology. Notwithstanding the empirical validation of the PoG and the PoP, an important shortcoming of this work lies in the omission of the eects to be observed from the principle of diminishing returns and the principle of labor division. Until they can be measured, the classical uni ed growth model cannot be said to have been fully con rmed by the data. We will therefore now turn closer to the rst equation of system 6.1. For future research, the most obvious extension of the classical uni ed growth model that has already been hinted at, would be to endogenize mortality. Here, it might be advisable to model a linear eect of a logged variable GDP per capita growth on death rate to account for an increasing di culty to generate a higher life expectancy.
Measuring the Neoclassical Theory of Production
In section 7.1 it was stated that a VAR(3) model is not able to capture the eects from the PoLD, since this principle can only be observed over a longer time frame than the suggested four years. In our simulations we have assumed that a larger birth rate will induce the corresponding cohort to increase GDP per capita only at one point in time exactly fteen years after they have entered the labor market. However, when trying to measure the operation of the PoLD in reality, we have to take the following two problems into account. Firstly, every individual tends to generate increasing per capita growth not only during the rst year of his working life, but over his complete lifetime due to the accumulation of human and physical capital. This means that the positive eect of population growth on GDP per capita is spread over the lifespan of the newborn individuals, which may even require the use of up to one hundred lags for empirical investigations.
Secondly, the PoDR operates at the same time continuously to mitigate the increase in GDP per capita. Consequently, the eects of these two principles the positive and the negative eects of population growth on GDP per capita growth are constantly intermingled in all our employed data. While we have seen that diminishing returns were signi cantly measured by the corresponding coirf of the simulation (left bottom graph of Figure 7.2), this eect could not be found in the stacked data sample (see left bottom graphs of Figure 7.4). Due to the inability of the VARapproach to estimate longrun relationships (which would require an enormous sample size to account for an enormous amount of parameters), we will in the following employ a simple growth regression approach to evaluate the neoclassical theory of production by measuring the PoLD and PoDR conjointly.
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8.1 Methodology
To support the advanced theory of production, the subsequent empirical exercise will focus on the estimation of steadystate equation 3.6, as stated in chapter three,
y∗t
yt−j∗ = bt−j
bt
1−γγ ,
and proceed as follows. To account for an appropriate j−value, we will in the following the-oretically justify a presumed time span between steady states. To nd evidence of a causal relationship of birth rate on GDP per capita growth, we will recover the remaining parameter γ by estimating the expression 1−γγ . Based on the assumption of constant returns to scale and indicating the supposed negative longrun eect of a change in the birth rate on productivity, the estimated value of the parameter γ is expected to lie within the range (0, 1). As a reference point, Cobb and Douglas (1928) estimated the production elasticity of capital of the (original) labor model to be α ≈ 0.25. More recently, Mankiw et al.'s (1992) estimations suggested a parameter value α ≈ 0.33 and most conventional calibrations assume a production elasticity of capital within the range (0.25, 0.33). Nonetheless, when withdrawing the variable human capital from the variable labor and adding it to the variable physical capital while treating the remaining
unskilled labor as population, Mankiw et al.'s estimator rises to α ≡ γ ≈ 0.66. Such an esti-mator would suggest a much higher exponent 1−γγ in equation 3.6 and correspondingly a larger leverage eect of changes in the birth rate on GDP per capita. In fact, if the following estimation con rms the conjecture that γ ≈ 0.66, population growth must be considered to have a much greater impact on economic development than is usually suspected.
Ideally, once we have estimated a consistent parameter value, we can con rm or reject the time frame presumption for j. To regress equation 3.6, we will employ the usual ordinary bivariate least squares (OLS) method. Due to the fact that the OLS estimator is the best linear unbiased estimator (BLUE), equation 3.6 will be linearized by taking logs of both ratios, yielding the approximate growth of GDP per capita as explained variable and the inverted approximate growth of birth rate as explanatory variable (see equation 8.1).1 OLS estimation is in this case a valid approach, since the variable birth rate is viewed as the (independent) source of all value in GDP per capita. This also implies that the additional use of an intercept or xed eects to account for unobserved eects is not necessary.
ln y∗t
However, when regressing equation 8.1 it must be noted that an ideal determination of β would only be realized under a comparison of the two steady states y∗tand yt−j∗ . Since the transition from one steady state to another might take a certain number of years after a shock in the variable birth rate has occured, we have to account for this transitional period by using an appropriate value for j. Although the birth rate is expected to aect productivity immediately negatively
1 For an alternative methodology including a measurement of convergence see Mankiw et al. (1992).
through bt, its positive eect of labor division is realized with a lifetimedelay through bt−j. To account for the latter eect, it has been stated that a newborn cohort will raise productivity only after it has generated the ability of unskilled labor with a maturity lag of one generation
of φ years, and it seems plausible to continue to assume a period of at least15 years. In addition, broad capital accumulation by way of dexterity and the invention of machines will be assumed to take place over the whole working life of a cohort, i.e. we add a maximum amount of ψ ≈ 50 years.2 Thus, since the above combined gains are probably fully achieved after j = φ + ψ years, we assume a maximum accumulation period of j ≈ 65 years to account for the transition between steady states. Consequently, ideal results from an OLS estimation can only be expected if we could employ time series of GDP per capita and birth rate over a time horizon of 2j ≈ 130 years where the birth rate stays constant for the rst j years, changing abruptly to another level (treatment) and remaining constant on the new level for another j years, as is exempli ed in Figure 8.1. After the birth rate has changed, GDP per capita is predicted to react positively over the latter period (treatment eect).
Figure 8.1: Illustration of two time series required for an ideal estimation of equation (11).
2 For a more extensive lag model, see e.g. Becker and Murphy (1992) or Liso et al. (2001).