## Introduction

## Issues in the Analysis of Malthusian Systems

_{t}is the natural logarithm of the real wage, b

_{t}is the crude birth rate, d

_{t}is the crude death rate, α

_{t}is the level of technology, and p

_{t}is the natural logarithm of population size. β is the elasticity of the real wage on population size; given diminishing returns of output with respect to population, its value is less than 1. In a standard Cobb-Douglas framework with two factors of production, of which one is fixed (e.g., land), (1 − β) is equal to the weight of labor in the production function and to the share of labor in total factor income, respectively. γ is the coefficient of the preventive check, and δ is the coefficient of the positive check. Finally, u

_{bt}and u

_{dt}model unsystematic exogenous shocks.

_{dt}in the form of an epidemic, such as plague, which drastically increases the death rate for a moment (Eq. (3)). Consequently, population declines (Eq. (4)) and the real wage increases (Eq. (1)). Through the Malthusian checks of Eqs. (2) and (3), the real wage increase translates into a higher birth rate and a lower death rate. As a result, population increases (Eq. (4)) and over successive periods drives population and the real wage back to their respective equilibrium values. Because the level of technology impacts on p* but not on w*, long-run technological change at low levels leads to an increase in population but not in the real wage (Eqs. (5) and (7)). This is the situation prevailing around the year 1500. Regions with high levels of agricultural technology were characterized by high population density but not by higher welfare levels in comparison with marginal regions (Ashraf and Galor 2011; Galor 2011: chapter 3). Subsequent real wage divergence (Allen 2001) may thus reflect differences with respect to the moment when economies transited from Malthusian to a post-Malthusian regime. Thus, an investigation into this transition holds the potential to provide relevant insights into the forces that produced modern economic development.

_{t}= α

_{t}− α

_{t − 1}. As long as g

_{t}remains sufficiently low, Malthusian adjustment fully compensates for technological progress. Consequently, the real wage and the vital rates are stationary around their equilibrium values given in Eqs. (5) and (6). In this case, Eqs. (1)–(4) can be analyzed with a VAR of the multivariate series (b

_{t}, d

_{t}, w

_{t}) (Eckstein et al. 1984; cf. Møller and Sharp 2014:113–115; Nicolini 2007).

_{t}has a unit root. If the influence of α

_{t}on the real wage prevails over the one of population in Eq. (1) either because of its magnitude or because of a weakening of the Malthusian checks, then the time series properties of α

_{t}translate to those of the real wage and the vital rates. Accordingly, these variables also have a unit root, and there are two cointegrating relationships in the system: namely, between the real wage and each of the vital rates. Because the preventive and the positive checks operate independently, there is no cointegration between the birth and the death rate (Møller and Sharp 2014:116–118; cf. also Bailey and Chambers 1993). In this situation, the estimation of structural time series models (Lee and Anderson 2002) and use of cointegrated VAR (Møller and Sharp 2014) are appropriate empirical strategies. Specifically, Møller and Sharp (2014:123–124), using the second method, showed that by the 1620s, the English economy was non-Malthusian in that the system of the vital rates was not stationary but included cointegrating relationships.

_{1}− c

_{2}. If technology is related to population size such that its equilibrium growth rate (g

^{∗}) is g

^{∗}= β(c

_{1}− c

_{2}), then technological progress compensates for the negative effect of population on the real wage. Consequently, stationary real wage and vital rates follow, and population is trend-stationary. Scale-dependent technology, which leads to an association between the level of technology and population size, is suggested by both Smithian and Boserupian views on economic development. Population growth increases market size, which encourages specialization along comparative advantage, thereby improving factor allocation and increasing the technical efficiency of the economy (Kelly 1997). In a Boserupian perspective, population growth facilitates the circulation of information and thereby contributes to the spread of labor-intensive technical innovations in agriculture (Boserup 1965). Our results suggest that during the half-century preceding national unification in 1871, Germany followed the pattern of a post-Malthusian regime with scale-dependent technology. Whereas a positive relationship between technology and population appears thus as a plausible ingredient of the transition out of a Malthusian regime, there is no theoretical justification for the condition g

^{∗}= β(c

_{1}− c

_{2}) to hold. If equilibrium occurs, it is fortuitous and potentially unstable.

_{t}is a vital rate (either b

_{t}or d

_{t},), w

_{t}is the natural logarithm of the real wage as earlier, R and L refer to lag orders, and u

_{t}is an i.i.d. error term; the remainder are parameters. Following the general logic of cointegration and ECM analysis, Eq. (8) produces estimates for the long-term and the short-term elasticities as well as the speed with which a deviation from the long-term relationship is corrected (cf. Brooks 2002:390–391; Greene 2012:959, 963–964; Lütkepohl 2007:246–247). The coefficient of the contemporaneous effect v

_{w0}indicates the central component of the short-term elasticity of a vital rate on the real wage (i.e., γ or δ in Eqs. (2) and (3)). Later in the article, we refer to v

_{w0}as the instantaneous elasticity of a vital rate on the real wage. In addition, we employ the sum of v

_{w0}and all v

_{w,r + 1}to characterize the cumulative short-term elasticity of a vital rate on the real wage. In substantive terms, the magnitude of the short-term elasticity of vital events on the real wage is a measure of the vulnerability or resilience of a population with respect to shocks regarding the availability of economic resources and thus constitutes an aspect of the standard of living (cf. Bengtsson et al. 2004). In a Malthusian perspective, a high short-term elasticity particularly of the death rate with respect to the real wage indicates a great importance of short-term shocks, particularly famines, for the adjustment of population to the means of subsistence (Malthus 1798/1998:43–44).

_{w}in the error correction term. It refers to a structural relationship between vital rates and the real wage that holds beyond short-term shocks and across several cohorts. The coefficient of the error correction term, λ, indicates the proportion of a deviation from long-term equilibrium that is corrected in one period—in this case, corrected within a year. It indicates the speed at which a vital rate adjusts to a shock in the real wage and thus provides a measure of how fast Malthusian adjustment takes place.

## Historical Overview and Data Sources

^{1}However, economic effects were considerable: the real wage reached its lowest level during the period under study in 1807, 1813, and 1817 (following the so-called Tambora crisis; Fig. 2). Postwar recovery and a series of bumper harvests in the 1820s, which benefitted workers’ living standards, quickly brought the real wage back to the level prevailing around the middle of the eighteenth century. Most remarkable of all is that it remained largely stable during the remainder of the period under study, despite an acceleration of population growth. Population could now expand at a rapid pace without a drastic fall in material conditions of life. Because this constitutes an important element of the post-Malthusian regime, the change of the relationship between the real wage and population growth gives a first indication that in Germany the Malthusian era ended in the 1810s.

### Properties of the (b, d, w) Series

Critical Values | ||||
---|---|---|---|---|

Null Hypothesis | Trace Statistic | 10% | 5% | 1% |

Multivariate Series (b, d, w) | ||||

1730–1870 | ||||

r = 0 | 87.02 | 39.06 | 42.44 | 48.45 |

r ≤ 1 | 34.86 | 22.76 | 25.32 | 30.45 |

r ≤ 2 | 7.10 | 10.49 | 12.25 | 16.26 |

1730–1799 | ||||

r = 0 | 58.50 | 39.06 | 42.44 | 48.45 |

r ≤ 1 | 24.07 | 22.76 | 25.32 | 30.45 |

r ≤ 2 | 8.50 | 10.49 | 12.25 | 16.26 |

1816–1870 | ||||

r = 0 | 59.29 | 7.52 | 9.24 | 12.97 |

r ≤ 1 | 31.29 | 17.85 | 19.96 | 24.60 |

r ≤ 2 | 9.55 | 7.52 | 9.24 | 12.97 |

Bivariate Series (b, w) | ||||

1730–1870 | ||||

r = 0 | 38.81 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 6.50 | 10.49 | 12.25 | 16.26 |

1730–1799 | ||||

r = 0 | 28.35 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 6.13 | 10.49 | 12.25 | 16.26 |

Bivariate Series (d, w) | ||||

1730–1870 | ||||

r = 0 | 54.09 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 14.59 | 10.49 | 12.25 | 16.26 |

1730–1799 | ||||

r = 0 | 38.66 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 12.16 | 10.49 | 12.25 | 16.26 |

Bivariate Series (b, d) | ||||

1730–1870 (lag order = 3) | ||||

r = 0 | 62.10 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 19.76 | 10.49 | 12.25 | 16.26 |

1730–1799 | ||||

r = 0 | 42.37 | 22.76 | 25.32 | 30.45 |

r ≤ 1 | 13.88 | 10.49 | 12.25 | 16.26 |

^{2}This is consistent with the fact that a stationary series cannot be part of a cointegrating relationship. The system (b, d, w) as a whole has a rank of 2 for the whole period under study and possibly also for the subperiod 1730–1799 (upper part of Table 1; for 1730, r ≤ 1 is rejected only at the 10% level of statistical significance). From the bivariate tests, it follows that one of the two cointegrating vectors captures the structural relationship between the birth rate and the death rate, whereas the other one is for the (stationary) death rate with zeros for all entries except for the death rate itself (Johansen 1995:37; Lütkepohl 2007:246, 250).

## Estimation of the Malthusian Checks

F | df1 | df2 | p | |
---|---|---|---|---|

1. H _{0}: b and d Do Not Granger-Cause w | 0.71 | 4 | 168 | .583 |

2. H _{0}: b Does Not Granger-Cause (d, w) | 0.80 | 4 | 168 | .523 |

3. H _{0}: d Does Not Granger-Cause (b, w) | 3.62 | 4 | 168 | .007 |

4. H _{0}: No Instantaneous Causality Between d and (b, w) | 9.03 | 2 | .011 |

Constant | Trend | w – 1 | ||
---|---|---|---|---|

Cointegrating Equation, Dependent Variable Is b – 1 ^{a} | ||||

Coefficient | 0.060 | –2.6E–05 | 0.005 | |

(1) | (2) | |||

Error Correction Equations, Dependent Variable Is Δb | ||||

ECT | –0.257* | (0.063) | –0.270* | (0.061) |

Δb – 1 | –0.232* | (0.077) | –0.237* | (0.077) |

Δd | –0.125* | (0.039) | ||

Δd – 1 | –0.102* | (0.039) | ||

Δw | 0.007* | (0.001) | 0.006* | (0.001) |

Δw – 1 | 0.010* | (0.002) | 0.008* | (0.002) |

R ^{2}, adjusted | .444 | .490 | ||

Durbin-Watson statistic | 1.90 | 1.94 | ||

Implied instantaneous elasticity | 0.19 | 0.16 | ||

Implied cumulative short-term elasticity | 0.45 | 0.37 |

1730–1799 | 1816–1870 | ||
---|---|---|---|

(1) | (2) | (3) | |

Constant | 0.083 ^{†} (0.043) | 0.015 (0.030) | 0.015 (0.012) |

Trend | –5.6E–05 (3.4E–05) | 2.0E–06 (2.3E–05) | |

d – 1 | 0.394* (0.112) | 0.469* (0.081) | 0.353* (0.146) |

d – 2 | 0.097 (0.112) | 0.249* (0.076) | 0.149 (0.145) |

w | –0.014* (0.006) | –0.013* (0.004) | 0.001 (0.003) |

w – 1 | –0.016 ^{†} (0.008) | 0.001 (0.006) | –0.001 (0.004) |

w – 2 | 0.023* (0.006) | 0.014* (0.004) | 0.000 (0.003) |

Year Dummy Variables | No | Yes | No |

R ^{2}, Adjusted | .503 | .795 | .102 |

Durbin-Watson Statistic | 1.97 | 1.97 | 2.02 |

Implied Instantaneous Elasticity | –0.41 | –0.38 | 0.02 |

Implied Cumulative Elasticity | –0.20 | 0.06 | 0.01 |

^{3}A recursive Chow test (but not cumulative sum (CUSUM) test) suggests structural breaks in 1808–1810 and 1814–1817, for which the F test statistic exceeds the critical value at the 5% level of statistical significance. This again lends support for the thesis that the beginning of the nineteenth century saw a major change in the nature and the strength of the Malthusian checks.

## Discussion

### The Preventive Check

### Unstable Malthusian Situation in the Eighteenth Century

### Transition to the Post-Malthusian Era in the Early Nineteenth Century

^{∗}, that equaled the constant rate of natural increase of population times the elasticity of the real wage on population, g

^{∗}= β(c

_{1}− c

_{2}) (cf. Issues in the Analysis of Malthusian Systems section). Thus, on the aggregate level, there apparently existed a linear relationship between the level of technology and population size; technological progress depended on changes in scale. Note that there is no theoretical reason why the equality g

^{∗}= β(c

_{1}− c

_{2}) should hold; at this point, the equilibrium we observe appears fortuitous.