The Solow Model
For long, the concept of economic change has been interesting economists. The reasons why some of the countries are poor while others are rich or even the understanding of why some nations grow faster than others has constantly disturbed the understanding of most economists. The theory of economic growth has been put upfront in explaining the growth indices of different nations. In its place, the theory of economic growth aims at explaining whether poor countries can grow faster than their rich counterparts, catch up with them and close their income gaps, that is converge. Elaborated by Trevor Swan and Robert Solow in 1956, the Solow model has been at the center of understanding the concept of economic growth since its inception. The model is a basic representation of simplified growth indices and assists in marking index differences between nations in specific periods. As such, Solow model bears the unique capability of comparing the growth rates of different nations. In their study, Solow and Swan assumed that the technological progress of a nation, saving rates as well as population growth rates contributes to the biggest percentage of the general growth in a nation. Based on a range of reputable sources, this paper aims at testing empirically how well the augmented and basic Solow models explains the phenomenon of convergence through an understanding of income variations across different nations between two different sets of periods. Ultimately, the study aims at investigating the impacts of proximate explanatory variables on income variation such as human capital, physical capital and the population.
Scope
The empirical study herein aims at covering 107 nations within two instantaneous periods of 1975 and the year 2009. The countries that are covered within the study detail 29 OECD nations, 30 nations from the sub-Saharan Africa, the Caribbean and Latin America contributes 21 nations as well as 27 nations from different geographical locations. The data that is focused for establishment of convergence detail the average real GDP growth rates over the specified periods, population growth rates, and the percentage of the GDP devoted to capital development and formation as well as the working age population that has attained the threshold level of education.
Notably, the conclusions of this paper could be biased. Based on the fact that the paper significantly ignores some other important aspects of economic growth such as government expenditures, rates of inflation, institutional or fundamental variables as well as private property rights are not included in the tabulations may, to some extent, trigger a sense of biasness. However, it is documented that the risk of multi-collinearity is highly enhanced with an input of wide ranges of variables in a study of such nature. In the paper, attempts to meet the convergence of different economies will be met through assumption of all the economies to be in their steady states. To establish the coefficients of each explanatory variable, regressions will be run through a range of variables with estimation of the differences taken into consideration. The convergence test is ultimately carried out with the nations assumed to be out of their states.
Theoretical Framework
The Solow model is a dynamic model aimed at explaining different growth avenues that nations exploit. Characteristically, the model is built upon neoclassical aggregate function:
Y (t) = [A (t), K (t), L (t)], where Y (t) represents the real outcome or the aggregate output at a particular time. The time is normally measured as a real GDP. K (t) represents the capital input in the function. The labor input is represented by L (t) while a measure of technology at a time or measure of productivity at a particular time is donated by a (t). As opposed to its predecessors, the Solow model bases its neoclassical character in the alternating proportions between labor inputs and capital inputs. Neoclassical theorists assert that the proportion between the aforementioned inputs can normally change based on the constantly changing prices of these two factors of production.
Like many other economic models, the Solow model assumes that an economy produces only one good. Such an economy should be motivated by perfection in competition both in the markets of the factors of production as well as in the markets of the goods. Generally, at a perfect competition status, the model assumes that such an economy is in the competitive general equilibrium. The aspect of competitive general equilibrium asserts that supplies of the final goods as well as the corresponding supplies of the factors of production are equal to their respective demand indices. In such a market, there are only two primary actors, which are the firms and the households. Further, all households as well as firms are considerably homogenous in such an economy.
Superficially, it is the households that own the factors of production. Such households supply labor in-elastically and it is the population of the nation that represents the labor supply. The population, and consequently the labor force grow at a constant rate, n. Firms outsource capital that they outsource from households. Ownership of capital from households is anchored on the continuous process of investments and savings that every households endeavor at placing at particular periodic frames. Solow assumed that households save sY fraction of their incomes at some particular time durations. The fraction of the income subjected to savings is referred to as the saving rate. In some economic quarters, the saving rate is donated by s. Notably, economies under the characterization of the Solow model are assumed to be closed. Thus, the amount saved (sY) is normally equal to the amount invested (I) at every particular time periods. Depreciation of the capital that household rents firms is denoted by δ and the rate is assumed to be constant. The net change in capital is efficiently described by the equation:
K̇=sY- δK,
Where the derivative of time is represented in the notation by the dot placed above K.
In the Solow model firms are driven purposely by the objective of profit maximization. As explained in the homogeneity aspect of the model, all the firms within the confines of the Solow model face similar production functions, thus, an assumption of a representative or aggregate production function is effected. Technology is known to root the basis of economic growth in most nations. In graphical presentations, it is the technology factor that shifts the production function during a presentation of a holistic Solow model. Generally, the factor of technology, A, incorporates the mechanisms through which factors of production get to be transformed effectively into the final outputs. Technology is non-excludable and non-rival, thus, the factor of technological change is presumably free. The non-rival aspect of technology is derived from the fact that application of changes in technology by one individual does not necessarily trigger usage by other individuals in similar contexts. Similarly, application of technology by an individual in the context does not deprive other individuals of using similar technology. The results of a technological change are universally usable, which makes technology characteristically non-excludable. From the illustrations of non-excludable and non-rival nature of technology, it is fair to conclude that technology, in an economy that runs in the confines of the Solow model is freely and largely available to all the firms within its confines and at a particular time frame. Finally, in such an economy, the technological factor I grow at a constant exogenous rate denoted by g.
Based on the assumptions of the Solow model discussed herein, the factor of technology and labor grow at constant exogenous rates of g and n respectively:
L(t)= L(0)ent
A(t)=A(0)egt
The aggregate production function used in the Solow model is neoclassical and is characterized by the diminishing marginal returns, labor as well as constant returns to scale. In cases where the production function exhibit constant returns to scale, escalation of factors of production increases the final input by similar proportions. An increase in one fundamental factor of production in the Solow model subsequently leads to an escalation of the total outputs, albeit through smaller proportions. Thus, the factor of production operates under the limitations of diminishing marginal returns in the model. The Cobb-Douglas production function normally employed to depict the neoclassical production function of the Solow model is thus:
Y (t) = A (t) K (t)α L (t)1- α , 0< α<1
Where the amount of effective labor is represented by A (t) L (t). normally, changes and advancements in technology are considered under some of the factors that make labor more effective in the Solow model. If A is represented as the primary factor that elevates the quality of labor, then growth rates can be met from calculating the income per unit of effective labor as a function of the capital per effective labor:
In the function, y represents the output per unit of effective labor while k represents capital per unit of effective labor. From the function, therefore, the main source of economic growth in the Solow model is rates of capital accumulation. The change in the capital accumulated within an economy leads to a subsequent change in the level of total income and is determined thus:
k̇ = sf (k) – (n + g+ δ)k, where sf (k) represents the fraction of the income that goes under savings from the households, effectively resulting into investments. From the function of the capital change, it is evident that level of investments and growth in capital are positively related. However, growth in capital is inversely proportional to depreciation rates, rates of technological changes as well as the growth rates of the population. Due to concavity of the investment curve and the diminishing marginal returns, the change in effective capital in the long run becomes zero per unit of effective worker. At the point where the capital ratio becomes constant and the capital does not accumulate anymore, the economy appears at its steady state and the income growth stabilizes. From the previous functions, the steady state condition can be derived from an analysis of the level of income thus:
with kss representing the steady state condition per unit of effective labor.
When the expression of steady state level of capital is substituted into a production function, the tabulation gives:
where, yss represents the output steady state level per unit of effective labor.
In 1992, MRW modified the Solow model by inserting the input of the human capital. In the insertion, human capital is described as all the qualities that humans may have to improve the quality of their productivity. Parameters such as health and education are considered relevant to human capital and consequently the growth of an economy. In their conclusions, MRW asserts that nations grow richer by having considerable amounts of both human capital and physical capital.
Through the concept of transition dynamics, the Solow model expounds the differences in growth rates of different nations. The transition dynamics states that nations that have the levels of income and capital below their determined steady states tend to grow faster than the growth rates within the steady states, which are the summation of the population growth rates relative to growth rates in technology. On the other hand, nations with income levels and capitals beyond their steady states tend to exhibit slow growth rates than their respective steady state growth rates. This hypothesis of convergence is popularly referred to as absolute convergence and whose tabulations are relevant both to the regressive and basic Solow models.
The following graphical representation depicts long term growth indices in the absolute convergence curves relative to the steady states of individual nations with transition dynamics in the classical models:
After a range of empirical studies, there have been a range of presentations indicating the presence of convergence among some nations and its absence among others. Effectively, different opinions have emerged behind convergence and the statistics behind its inconsistent nature. More reputable of the refuting assertions is the fact that nations differ with respect technological growth rates, population growth rates and investment rates. Thus, such nations exhibit different steady states almost unconditionally. The countering neoclassical models state that convergence can only be evident among nations with similar steady states, which is not always the case.
Presentations
In reaching out to its conclusions, this paper explores descriptive statistics. Thus, the economic behaviors of different nations through the parameters of position are calculated. In each sample of the Solow model, position measures of mean, mode and median as well as measures of dispersion such as coefficients of variations, standard deviations among others are employed in developing a correlation between growth rates in different economies via the lens of convergence. The observed nations detailed Sub-Sahara Africa whose convergence diversifications are highly explained through their diversity in building physical capital. Countries within Sub-Saharan Africa detail a significant similarity in population growth rates, changes in technology as well as capital stock indices, making them highly effective in the regressions of steady states, thus convergence indices. OECD group of nations is the second set in the study, which statistically records 1.5% with 55% of CV within the study period of 1975-2009. Latin nations exhibit varied population and economic growth rates. Besides, significant similarities in Latin nations are evident in school enrolment rates and capital formation rates. Mixed nations in the study evidently offer a difference in real GDP per worker as well as different expansion rates. Mixed nations show a slight similarity development of stock capital.
The following tables depict the correlation matrices of the chosen nations.
The coefficients on the natural logarithm of capital and population growth are equal in absolute values. Given that α, the share of capital in the total income is expected to roughly be 1/3, the regression coefficients for the two explanatory variables should then approximately be equal to 0.5 and -0. 5. Estimations are made both with and without imposing restriction on the coefficients. The results of which depict the following unconditional convergence rails across different economic regions.
The above graphical presentations exhibit the absence or presence of unconditional convergence. In the presentations, GDP/ worker 1975 (in the x-axis) is plotted against the per worker income growth rates for the periods falling between 1975 and 2009 for different economic regions. The possibility of convergence is inferred from observation of a downward slopping trend from left to right of the graphs.
Observing the graphs above, we notice that only OECD countries show a tendency to converge to the same steady-state level of per worker income in the long run. These countries are similar in n, s and g. This is probably the reason why the study obtains a very high fit in the particular sample. The results presented in the graphs above confirm the findings of different studies indicating that among very similar countries, like in the OECD, the convergence will be clearly observed. However, for a wider group of countries a more general model seems to be required, like the model of conditional convergence.
Conclusion
With the aim of the study anchored upon examination of convergence among different nations, it is established that absolute and conditional convergence are clearly evident in regional settings of each set of data. Among the OECD nations, absolute convergence is evident and the convergence attributed to their similarity in functions such as technological growth rates, similar rates in capita depreciation and similar population growth rates. In the augmented models, it is further evident that the OECD nations exhibit an overall fit on the initial levels of incomes and income differences. Simply, absolute convergence will only be realized within countries with similar attributes of the Solow model: population growth rates, investment patterns or changes in technology. Otherwise, nations with different attributes may only be subject to conditional convergence out of their individual springs towards or out of their steady states. Either way the Solow model is an absolute show of how different nations grow towards or away from each other. As indicated, the poor nations may be having faster growth rates, however, their rates of diminishing returns may be limiting them from bridging the gaps between their economies and those of the wealthy nations.
References
Gamber, E., Colander, D.C (2006). Macroeconomics. Cape Town: Prentice Hall
Karabona, P., & Koutun, A. (2013). An Empirical Study of the Solow Growth Model. Malardalen University Press
Lepel, C. F (2007). The Solow Model. Munchen GRIN Verlag
Solow, M. R. (2016).A Contribution to the Theory of Economic Growth: The Quarterly Journal of Economics, Vol. 70, No. 1 (Feb., 1956), pp. 65-94.
