Introduction

            Mathematical models refer to descriptions of systems using mathematical language and concepts. Mathematical modelling refers to a process of creating a mathematical model with the goal of analysing real life situations (Acemoglu 2008). Models are helpful in various disciplines such as natural sciences and engineering. Social sciences such as psychology, economics, political sciences, and statistics among others also use mathematical models extensively. Models can help in the study and revealing of the effects of various components of a system. 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). The period that exhibited no changes in living standards included both Malthusian and Post-Malthusian 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.

Figure 1: income per capita as projected by the Malthusian model

            Malthusian model with no technological changes or capital uses N₀ to denote the initial number of people alive and N_t 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 people consumed much food they would grow stronger and healthy; hence, their bodies would fight diseases (Thompson, 2011). The graphical representation below summarizes the above information.

Figure 2: Birth and death rate as functions of per capita consumption (Caselli et al 2006)

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            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 = N_t. 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 (c_t). 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 Y_t, the economy uses labour and land to produce a single final good. The production function, Y_t =AL^?_t N _t^1-?. 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-? represents share of labour. Constant returns to characterize the production function, which increases at each of the two inputs that depend on 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 N_t 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References

Acemoglu, D 2008, Introduction to Modern Economic Growth, 3rd edn, Princeton University Press, Princeton.

Caselli, G, Vallin, J & Wunsch, G 2006, Demography: Analysis and Synthesis, Volume 1, 2nd edn, Academic Press, Boston.

Clark, G 2008, A Farewell to Alms: A Brief Economic History of the World, 2nd edn, Princeton University Press, Princeton.

Crafts, N & Mills, C 2009, ‘From Malthus to Solow: How Did the Malthusian Economy Really Evolve?’, Journal of Macroeconomics, vol 31, no. 1, pp. 68-93.

Hansen, D & Prescott, E 2002, ‘ Malthus to Solow’, American Economic Review, vol 92, no. 4, p. 1205–1217.

Thompson, J 2011, Empirical Model Building: Data, Models, and Reality, 2nd edn, John Wiley & Sons, Hoboken.