Mathematical investigations of the escape from the Malthusian trap

"We present a simulation model that synthesizes Malthusian and Boserupian notions of the way population growth and economic development were intertwined. The non-linear stochastic model consists of a system of equations whose dynamics culminate in an industrial revolution after hundreds of iterations. The Industrial Revolution [in Europe] can thus be conceptualized as a permanent 'escape' from the Malthusian trap that occurs once the economy is capable of permanently sustaining an ever growing population. We investigate the conditions for such an escape and their sensitivity to the parameters of the model....Our results show that the likelihood of an escape is sensitive to the savings rate and to the output elasticities of the two sectors of the economy. When not in a subsistence crisis, the chances that an escape will occur increase for larger values of the ratio of the savings rate to the growth rate of the population. The chances of an escape also increase substantially for larger values of the output elasticities of labor." (SUMMARY IN FRE)     

A model on the escape from the Malthusian trap

We consider a demoeconomic model where output is produced using physical capital, human capital and technology as inputs. Human capital depends on the number of people and the level of education in the economy. The dynamics of labour, physical capital, education and technology are endogenously determined such as to reflect the interdependence between economic and demographic factors. The longrun path of the economy and in particular the possibility to escape the Malthusian trap crucially depend on technological progress, which provides for economy wide increasing returns to scale. The build up of technology is positively related to the stock of human capital. Our model predicts that positive population growth is sufficient to escape the Malthusian trap. Received: 22 August 1996 / Accepted: 11 December 1997     

Human capital,technological progress and the demographic transition*

We emphasize the importance to consider components of population growth — fertility and mortality ‐ separately, when modeling the mutual interaction between population and economic growth. Our model implies that two countries with the same population growth will not converge towards the same level of per capita income. The country with the lower level of birth and death rates will be better off in the long run. Introducing a spill over effect of average human capital on total productivity our model implies multiple equilibria as illustrated in Becker el al. (1990) and Strulik (1999). Besides the existence of a low and high level equilibrium ‐ as characterized by low and high levels of per capita output respectively ‐ we show the existence of multiple low level (Malthusian) equilibria. Initial conditions and parameters of technological progress and human capital investment determine whether an economy is capable to escape the low level equilibrium trap and to enjoy sustained economic growth.     

A stochastic version of the malthusian trap model: Consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross‐sectional correlation between these two growth rates under two additional assumptions: i) the relations in the model at national levels include country‐specific and time‐invariant random components, and ii) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these two assumptions can explain near‐zero correlations between the two growth rates even if there exist a strongly negative effect of population growth on economic growth. However it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

Induced population growth and induced technological progress: Their interaction in the accelerating stage

A simple model of Malthusian population growth combined with population‐induced technological progress generates accelerating growth. The model may be relevant for a first stage of growth in which natural resource limitations can be overcome through technological progress; it is not applicable to a later stage in which resource constraints are more resistant. Parameter values are roughly inferred from historical experience. Exogenously more rapid population growth initially depresses income, perhaps for up to several centuries, then raises it without limit. More rapid population growth is desirable only when the social discount rate is less than the ratio of the parameters for induced technical progress and static diminishing returns. Imposed population fluctuations cause inverse movements in incomes, so that induced progress is very difficult to detect empirically even for population fluctuations up to 500 years.     

On demographic transition,structural change,and economic growth and stagnation

The paper analyzes an economy with an agrarian and an industrial sector. Demand is determined by Engel's Law. Population growth follows a non‐linear income dependent path according to the theory of demographic transition. In case of decreasing returns to scale in the agrarian sector the existence of a stable low‐income equilibrium with high population growth can be shown. If this equilibrium is globally unstable, the system evolves towards a steady‐state of perpetual economic growth and low population growth. The path of demographic transition coincides with a path of structural change from an economy specialized in agriculture to a fully industrialized economy. The introduction of an income dependent savings rate allows the interpretation of the low‐income equilibrium as a limit cycle and, therefore, the explanation of high fluctuations in population growth and per capita income in least developed economies.     

Malthusian Dynamics in a Diverging Europe: Northern Italy, 1650–1881

Recent empirical research questions the validity of using Malthusian theory in preindustrial England. Using real wage and vital rate data for the years 1650–1881, I provide empirical estimates for a different region: Northern Italy. The empirical methodology is theoretically underpinned by a simple Malthusian model, in which population, real wages, and vital rates are determined endogenously. My findings strongly support the existence of a Malthusian economy wherein population growth decreased living standards, which in turn influenced vital rates. However, these results also demonstrate how the system is best characterized as one of weak homeostasis. Furthermore, there is no evidence of Boserupian effects given that increases in population failed to spur any sustained technological progress.     

A stochastic version of the Malthusian Trap Model: consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross-sectional correlation between these 2 growth rates under 2 additional assumptions: 1) the relations in the model at national levels include country-specific and time-invariant random components, and 2) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these 2 assumptions can explain near-0 correlations between the 2 growth rates even if there exist a strongly negative effect of population growth on economic growth. However, it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

Population, population density and technological change

In a model on population and endogenous technological change, Kremer (1993) combines a short-run Malthusian scenario where income determines the population that can be sustained, with the Boserupian insight that greater population spurs technological change and can therefore lift a country out of its Malthusian trap. We show that a more realistic version of the model, which combines population and population density, allows deeper insights into these processes. The incorporation of population density also allows a superior interpretation of the empirical regularities between the level of population, population density, population growth, and economic development at aggregated and disaggregated levels.

不同预测模型下西藏自治区人口发展状况及对策研究

针对西藏自治区人口数据的有限性、数据序列的不平滑性等特点,分别采用一元回归、马尔萨斯模型、lo-gistic模型、GM(1,1)模型等4种方法,利用西藏自治区1980—2009年统计年鉴数据进行人口预测,综合考虑各种方案预测值,确定西藏自治区2010-2030年的人口数,在此基础上探讨基于人口增长的西藏经济发展模式与对策。结果表明,几种预测模型中平均相对误差率最小的是一元线性预测模型Ⅲ,仅0.2611%,马尔萨斯预测模型Ⅲ的平均相对误差率最大,也只有2.0767%;从预测结果看,Logistic模型的预测结果最为保守,仅为339.61万人,马尔萨斯模型预测最为乐观,高达380.10万人,未来相关研究中,可应用Logistic模型预测值作为下限,马尔萨斯模型预测值为上限;为综合应用四种模型的优势,研究中综合平均四个不同模型的预测结果,则西藏2010年人口将达到294.07万人,2015年达到311.34万人,2020年328.81万人,2025年346.54万人,2030年364.57万人,未来20年总人口增长趋势逐渐加快,区域发展中有一定人口压力。     

Population aging, health care, and growth: a comment on the effects of capital accumulation

In a recent paper, Hashimoto and Tabata (J Popul Econ 23:571–593, 2010) present a theoretical model in which the increase in the rate of dependence due to aging of the population leads to a reallocation of labor from non-health to health production and, as a consequence, to a decline in economic growth. We argue that these results rely heavily on assumptions of a “small economy” and perfect capital mobility, which tie down the amount of capital. In this paper, we proceed by analyzing the case of an economy in which the availability of capital is endogenously determined by domestic savings. We find that the new “capital accumulation effect” is opposite to the previous “dependency rate effect,” leaving the effect on economic growth ambiguous. In particular, if the former prevailed, population aging would foster economic growth, a result that finds support in recent empirical work.     

Induced population growth and induced technological progress: their interaction in the accelerating stage

"A simple model of Malthusian population growth combined with population-induced technological progress generates accelerating growth. The model may be relevant for a first stage of growth in which natural resource limitations can be overcome through technological progress; it is not applicable to a later stage in which resource constraints are more resistant. Parameter values are roughly inferred from historical experience." (SUMMARY IN FRE)     

There is no low-level fertility and development trap

Summary The theory of the low-level equilibrium trap asserts that an increase in income stimulates population growth sufficiently so that the additional people 'eat up' the 'surplus' over subsistence, and hence drive the level of income back to subsistence. Originally the theory referred primarily to mortality, but nowadays its application is to fertility. In the long-run equilibrium context in which the theory is ordinarily presented, the fact that the long-run elasticity of fertility with respect to income is negative in less developed countries fatally contradicts the accepted version of the trap. But to give every chance for trap theory to be meaningful, the paper presents a period-by-period analysis, embodying larger-than-observed positive elasticities during the early years and the logically necessary counterbalancing negative elasticities during the later years. These elasticities are combined with consumption and production figures for various age groups to estimate the effect in each year after the windfall, and altogether. The results show that even under assumptions not charitable to the conclusion of this paper, additional children do not even come close to 'eating up' the increase in income which induced their births, so that the trap theory is falsified.     

There is no low-level fertility and development trap

The theory of the low-level equilibrium trap asserts that an increase in income stimulates population growth sufficiently so that the additional people ‘eat up’ the ‘surplus’ over subsistence, and hence drive the level of income back to subsistence. Originally the theory referred primarily to mortality, but nowadays its application is to fertility. In the long-run equilibrium context in which the theory is ordinarily presented, the fact that the long-run elasticity of fertility with respect to income is negative in less developed countries fatally contradicts the accepted version of the trap. But to give every chance for trap theory to be meaningful, the paper presents a period-by-period analysis, embodying larger-than-observed positive elasticities during the early years and the logically necessary counterbalancing negative elasticities during the later years. These elasticities are combined with consumption and production figures for various age groups to estimate the effect in each year after the windfall, and altogether. The results show that even under assumptions not charitable to the conclusion of this paper, additional children do not even come close to ‘eating up’ the increase in income which induced their births, so that the trap theory is falsified.     

中国城镇居民家庭储蓄水平队列效应分析

利用中国家庭收入调查（ CHIP）数据对城镇居民家庭的储蓄率进行了队列效应的实证研究。分析结果表明,属于不同队列的城镇家庭在储蓄水平上都呈现出相似的“U”型年龄分布,即中年家庭（40－50岁）储蓄率低,年轻家庭（20－40岁）和退休家庭储蓄率较高,这一现象与其他国家的实证研究结果有显著不同。本文认为,中国城镇居民储蓄率的“U”型年龄分布主要是由人口与经济的高速增长带来的：人口高速增长带来的队列规模效应使得面对更大竞争压力的年轻人口储蓄率较高,而经济水平持续增长带来的消费水平增长预期使得接近退休家庭储蓄水平重新上升。     

The optimum growth rate for population reconsidered

This article gives exact general conditions for the existence of an interior optimum growth rate for population in the neoclassical two-generations-overlapping model. In an economy where high (low) growth rates of population lead to a growth path that is efficient (inefficient), there always exists an interior optimum growth rate for population. In all other cases, there exists no interior optimum. The Serendipity Theorem, however, does, in general, not hold in an economy with government debt. Moreover, the growth rate for population that leads an economy with debt to a golden rule allocation can never be optimal.      

Population: Past growth and future control

This paper is the opening chapter and demographic context of a forthcoming book onThe World's Food. It offers a summation of the Malthusian perception of food and population, an overview of population growth in history and the role of the demographic transition. The varying experiences of developing countries in bringing down the birth rate are examined and used as a basis for projecting future population growth.     

On the relationship between population and means of subsistence]

Population is the basis of all social activites and social production. Population growth and development must have a definite means of subsistence to meet its cultural and material needs. The larger the population of a country, the greater is its demand for consumer goods and, likewise, the yield of its means of subsistence should be greater. Population brings about the unification of production and consumption. Furthermore, the ratio of population to the means of subsistence must be maintained at an appropriate level. Population growth must be slower then the growth of the means of subsistence in order to ensure continuous economic expansion and population increase. However, there are some people, notably Malthus, who believe that the balance between population growth and the means of subsistence should be equal. It is crucial to note differences between Marxist and Malthusian points of view. The basic outlook on the nature of the relationship between population and the means of subsistence is different. For Malthusians, it is a matter of the number of people and the quantity of the means of subsistence. For Marxists, the relationship is a historically determined social relationship. For Malthusians, population development is the primary force behind social development, i.e., the imbalance between population and the means of subsistence stems from social ills. Marxists differ from this in believing that population cannot be divorced from material production. Malthusians believe that population surplus derives from a population increase that is greater than an increase in the means of subsistence. Marxists believe a population surplus is also an historically determined social relationship. The Malthusian outlook for the future of population and the means of subsistence is pessimistic, whereas the Marxian view embodies the optimism of revolution.     

Malthus and the Philippines

Jose D. Drilon, Jr., president of Food Terminal, Inc., and a former undersecretary of the then Department of Agriculture and Natural Resources in the Philippines, attributes the widening gap between food supply and demand in developing countries to the high rate of population growth and to the inability of poor countries to produce more food. This situation, in which many countries are facing hunger, was predicted by Thomas Robert Malthus as early as the 16th century. The primary concern of Malthus was the problem of making the food supply keep pace with a constantly growing population. The question arises as to how reliable is Malthusian theory. According to Drilon, Malthus was correct in predicting that population would expand at a rate not previously imagined but that the other aspects of Malthusian theory might not hold true due to the intervention of human beings. For example, it is hoped that the imbalance between population growth and food production can be minimized in the future. In the Philippines there is good reason to be concerned about the validity of Malthusian theory. Although the country's growth rate has been reduced from 3.01% in 1970 to 2.6%, it is still quite high. However, the Philippines has actually been producing sufficient food to feed its population. To make the Philippines self-sufficient in rice, the government initiated the Masagana 99 program in May 1973. Technical and material resources from the public and private sectors were provided to aid rice producers. A nationwide information campaign was also launched to familiarize the farmers with the new methods of rice culture. Masagana 99 has been costly but effective. Since the launching of the program, rice production in the Philippines has been increasing at 7% a year. The government is now using the Masagana 99 formula to increase the production of other crops.     

The Relation Between Actual and Intrinsic Growth Rates

This paper demonstrates that, to a close approximation, the intrinsic growth rate of a population is equal to the mean of age-specific growth rates below age T, the mean length of a generation. This mean is normally close to the growth rate of the entire population block below age T. Therefore, when a disparity exists between the intrinsic growth rate and the actual growth rate of a population (whether or not net migration is included in both rates), it must be attributable to an unusual growth rate of the population block above age T. One implication is that the ‘momentum of population growth’ is entirely confined to the age span above T, approximately age 28 in developing countries to-day.