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Malthusian Model 1
MALTHUSIAN MODEL
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Malthusian Model 2
Malthusian Model
Introduction
Mathematical models refer to descriptions of systems by using mathematical language
and concepts. Mathematical modelling refers to a process of creating a mathematical model
(Acemoglu, 2008). Models are helpful in various disciplines such as natural sciences and
engineering. Social sciences such as psychology, economics, political sciences, and statistician
among others also use mathematical models extensively. Models can help study and reveal the
effects of various components. They also assist in making predictions about behaviour of a
system. They take various forms such as, and not limited to, statistical models, dynamic systems
and differential equations (Clark, 2008). In relation to this, this study discusses the Malthusian
model, also referred to as simple exponential growth model.
The Malthusian model comprises of two primary components. The first component is a
production function, which has a predetermined or fixed factor of production. The term “fixed or
predetermined” implies that the supply cannot change over a given period (Clark, 2008). The
Malthusian model takes land as fixed factor since both capital and labour can increase over time.
The second component of the model is population growth function, which is an increasing factor
of consumption of per capita. The two components guarantee a steady state characterized by
steady living standard even when there are technological dynamics.
The study first proceeds to study the model with the absent of technology and capital to
assist in creating a perception for the model. The study achieves this by using the graph below
(Caselli et al., 2006). The graph is a plot of per capita income against time from 3000 BC to 2000
AD. According to the graph, the living standards showed no changes for the first 4800 years. The
standards of living exploded after 1800 AD, which marked the modern regime (Clark, 2008).
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The period that exhibited no changes in living standards included both Malthusian and PostMalthusian regimes. One can conclude that the living standards remain constant because there
were no technological changes and the interplay of various factors present today. The perception
created by the model is that a system cannot change without the interplay of factors.
Technological changes might have resulted in increase in living standards after 1800 AD.
Malthusian model with no technological changes or capital uses N0 to denote the initial
number of people alive and Nt to denote the number of people in an economy at certain date t. In
relation to demographics, the model points out that death rate and birth rate of a population
determines its growth. Malthus advanced a population dynamics theory, which stated that the
living standards did not determine the birth rate of a population. However, the theory perceived
death rate as a decreasing function of the amount of the per capita consumed. This perception
originates from the notion that if consumed much food they would grow stronger and healthy;
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hence, their bodies would fight diseases (Thompson, 2011). The graphical representation below
summarizes the above information.
.
Birth and death rate as functions of per capita consumption (Caselli et al., 2006)
The growth rate of a population for a certain level of per capita consumption is equivalent
to the difference between death rate and birth rate (Hansen & Prescott, 2002). Malthusian
denoted this function as g(c). There is no population growth when both birth rate and death rate
are equal, denoted as N t+1 = Nt. This formula can be simplified to N t+1 = N t [1+g(c)]. The gross
population growth denoted as G(c) is equivalent to 1+g(c), which implies that N t+1 = N t G (ct).
According to the model, everyone in the economy is gifted with one unit of time that he or she
can use to work (Hansen & Prescott, 2002). However, land is fixed; hence, all people have an
equal share of land. Land does not undergo any depreciation.
In relation to production function denoted as Yt, the economy uses labour and land to
produce a single final good. The production function, Yt =ALαt N t1-α. A represents the total factor
of productivity (TFP) (Crafts & Mills, 2009). This function of production is almost similar to
Solow model’s production except that 1-α represents share of labour. Constant returns to
characterize the production function, which increases at each of the two inputs that depend on
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fixed land. In addition, the law of diminishing return is applicable to every factor independently.
The increases in output linked to an extra unit of labour input diminish as labour increases while
land is fixed. It is important to note that land is crucial in production.
The Malthusian model determines equilibrium quantities of three markets, which are
labour market, goods market and land rental market (Crafts & Mills, 2009). The supply of both
land and labour are both since individuals supply their entire land and time endowment to the
market. Since Nt and L are the equilibrium inputs of the system, it is inconsequential to
determine amount of output at equilibrium point.
The two components of the Malthusian model have an association (Acemoglu, 2008).
Notably, sustained growth in consumption per capita output cannot be realized because the
increasing the amount of land used by an individual is the only way of increasing his or her
productivity. However, the fixed nature of land, as accorded by the model, makes it difficult to
increase. Constant path of per capita output and per capita consumption characterizes a steady
state of equilibrium in the system (Clark, 2008). At equilibrium, population growth is zero. This
is the reason why steady states have constant populations. if not, with the law of diminishing
returns, per capita consumption and output reduces, forcing the system to add more individuals
to work on land.
In conclusion, Malthusian model addresses economic and demographic issues. The
population dynamics theory states that the living standards did not determine the birth rate of a
population. The economy uses labour and land to produce a single final good. Constant returns to
characterize the production function, which increases at each of the two inputs that depend on
fixed land.
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References
Acemoglu, D., 2008. Introduction to Modern Economic Growth. 3rd ed. Princeton: Princeton
University Press.
Caselli, G., Vallin, J. & Wunsch, G., 2006. Demography: Analysis and Synthesis, Volume 1. 2nd
ed. Boston: Academic Press.
Clark, G., 2008. A Farewell to Alms: A Brief Economic History of the World. 2nd ed. Princeton
: Princeton University Press.
Crafts, N. & Mills, C., 2009. From Malthus to Solow: How Did the Malthusian Economy Really
Evolve? Journal of Macroeconomics, 31(1), pp.68-93.
Hansen, D. & Prescott, E., 2002. Malthus to Solow. American Economic Review, 92(4), p.1205
1217.
Thompson, J., 2011. Empirical Model Building: Data, Models, and Reality. 2nd ed. Hoboken:
John Wiley & Sons.
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