Solow Model

What are the main limitations of the Solow model? Discuss with reference to theory and evidence.

The Solow Model, also known as the neoclassical growth model or exogenous growth model is a neoclassical attempt created in the mid twentieth century, to explain long run economic growth by examining productivity, technological progress, capital accumulation and population growth. This model was contributed to by the works of Robert Solow, in his essay ‘A Contribution to the Theory of Economic Growth’ and by Trevor Swan in his work, ‘Economic Growth and Capital Accumulation’, both published in 1956. The model is perceived to be an extension of the 1946 Harrod-Domar model, which Solow (1956) describes as a ‘model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions.’

Instead Solow (1956) supposes that a ‘single composite commodity is produced by labour and capital under the standard neoclassical conditions.’ This notion will be elaborated on further in the course of this essay. Economists today use the Solow model source of growth accounting to estimate the individual effects on economic growth of capital, labour and technological change and thus the model is contemporarily significant. This essay aims to first and foremost outline the assumptions and features of the Solow Model and explain them. It will follow with a discussion of the determinants of economic growth within the model and highlight the limitations within the framework of the model. Finally, this essay will evaluate the effectiveness of the model in determining long-term growth.

Assumptions:

In this model, Solow makes a number of assumptions that have been critiqued by many theorists as being limitations of the model it self. The first of these is that the model is constructed amidst a one good economy and the good is homogenous in nature. As there is only one good produced, consumed and invested in, the economy is abstracted from any trade.

The model also tends to ignore the demand side of production therefore the model assumes output (Y) is a function of labour (L) and capital (K):

Y= F (K, L)

However, there are two key assumptions of the production function crucial to the structure of the model. The first assumption is that the production function exhibits constant returns to scale for capital and effective labour, i.e. a doubling of inputs leads to a doubling of output). In the production function below, this assumption can be mathematically proven by the ‘1-+.’

Y = F [K, L] = Kα L1-α

This assumption itself can too be segmented into two separate assumptions about the economy, first being, gains from specialisation have a minimal impact on the economy. Secondly, inputs such as land or natural resources are nonessential (Romer, 2006, pg.10). Romer (2006) warrants this consideration on the basis that in practice, natural resources do not tend to prove as a constraint on growth.

A second assumption of the production function is that there are diminishing returns to each factor individually; decreasing marginal returns to factor accumulation (adding extra capital with labour being constant yields ever-smaller increases in output):

Henceforth, the flattening of the graph below, illustrates the phenomenon, diminishing returns to capital, which Jones explains to as, ‘each additional unit of capital we give to a single worker increases the output of that worker by less and less’ (Jones, 1998, pg.21).

Fig. 1

Another key assumption of the model is that all savings are invested,

S = I = s. Y

Therefore, the savings rate is constant and thus is the investment rate; this notion will be described below when discussing the structure of the model.

As Robert Solow was a neoclassical economist, some assumptions common to the school of thought apply to this model as well. These include perfect competition in the labour and capital markets, hence assuming that perfect competition reigns in all markets (Harcourt, 2006, pg. 111). Furthermore, two of the more disputed assumptions made by the model are, labour force grows at rate n, which is constant and exogenously determined. Also, technology is regarded to be exogenous and is not explained by the model. Both these assumptions have been used by many economists to critique the model and contribute to the limitations of the model elaborated on further in the essay.

What is the Solow Model designed to show?

The Solow Model is designed to show how the growth in the labour force, capital stock and advances in technology interact and how they affect a nations total output. The model is important for the analysis of economic growth in developing countries as it demonstrates the nature of an economy to be a key determinant of steady-state capital stock within a country. If the savings rate is high, the economy will have a large capital stock and thus high level of output and vice versa. Correspondingly changes in capital stock can lead to economic growth.

The Solow model indicates that countries with high population growth (with no change in capital) will have lower levels of output per person. In the model therefore, population growth capital per worker and output per worker are constant. Correspondingly, the aim of the Solow Model becomes clear: it is to show that an economy will incline towards a long-run equilibrium K/L (k) ratio at which Y/L (y) is also in equilibrium, so that Y, K and L all grow at the same rate, that is n. Ultimately the model predicts long run equilibrium at the natural rate of employment.

Y= A. F(K, L, H)

Where Y is output, A is total factor productivity, K is capital stock, L is labour stock, and H is human capital.

In this Cobb-Douglas production function, α is the elasticity of Y with respect to K, and (1-α) is the elasticity of Y with respect to L.

Therefore, if we rearrange the function in respect to one unit of labour, we will end up with the equation below where y is Y/L and k is K/L.

The diagrammatic illustration of the Solow Model is shown below.

Fig. 2

Another way of explaining this curve is states that the rate of change of capital stock per unit labour is the difference between actual investment and Break-even investment shown in the diagram below.

Fig.3

The two lines cross at k* where both investments are equal.

Determining Economic Growth in the Solow Model:
The first part of this essay has introduced the Solow Model and set out the assumptions of the model and its diagrammatic representation. This essay will now focus on the determinants of economic growth within the model and then outline the limitations of the model whilst drawing parallels between the two. The essay will also refer to the assumptions of the model it self when discussing its limitations.

The first reason for growth in the model is when an economy is in disequilibrium i.e. the supply of capital is not equal to the demand for capital. The first part of this essay has introduced the Solow Model and set out the assumptions of the model and its diagrammatic representation. This essay will now focus on the determinants of economic growth within the model and then outline the limitations of the model whilst drawing parallels between the two. The essay will also refer to the assumptions of the model it self when discussing its limitations.

The first reason for growth in the model is when an economy is in disequilibrium i.e. the supply of capital is not equal to the demand for capital.

Fig. 4

The point to the right and left of k* (equilibrium capital-output ratio) are in equilibrium. If s.f k>n+δ k then the economy is growing towards the equilibrium and the growth rate is positive. Examples of countries moving towards equilibrium are China, India and other BRIC countries, which have experiences high levels of growth in the last decade but are now slowing down. Other examples include Japan and Germany that had their capital stocks exhausted by World War II, but have grown more rapidly than the US in the last fifty years (Jones, 1998, pg. 40). If s.f k<n+δ k then the economy is shrinking towards the equilibrium and are experiencing negative growth. The point to the right and left of k* (equilibrium capital-output ratio) are in equilibrium. If s.f k>n+δ k then the economy is growing towards the equilibrium and the growth rate is positive. Examples of countries moving towards equilibrium are China, India and other BRIC countries, which have experiences high levels of growth in the last decade but are now slowing down. Other examples include Japan and Germany that had their capital stocks exhausted by World War II, but have grown more rapidly than the US in the last fifty years (Jones, 1998, pg. 40). If s.f k<n+δ k then the economy is shrinking towards the equilibrium and are experiencing negative growth.

A second reason for growth within the model is a response to an increase in the savings rate, causing a change in equilibrium from k’* to k’** due to the upward shift in savings from s y’ to s y’’, illustrated below:

Fig. 5

However, it is to be noted that an increase in the savings rate has a short-term effect on the growth but does not cause long run growth. Moreover, the assumption of diminishing returns to capital mentioned previously implies a limited prospect for achieving growth via this means.

A third means of growth in the Solow Model is an increase in technical progress, shown in the diagram below, as a result in an upward shift from y= f (k) to y= f’ (k) and an increase in the capital output ratio from k* to k**.

Fig. 6

Arguably, technical progress is assumed to be the only explanation for persistent increases in per capita growth, however, the fact that it is determined exogenously within the model it self proves to be a predominant flaw in the framework of the model and is criticised by many of Solow’s neoclassical predecessors. This argument will be discussed as a limitation of the model in the next part of this essay.

Limitations of the Solow Model:

As mentioned previously, the most prevalent of limitations of the Solow Model is its assumption that technical progress is exogenous, exacerbated by the fact that the model considers it to be a means of growth. Furthermore, the models answer to why countries exhibit sustained growth is also technical progress; without technical progress, there is not much scope for per capita growth as a result of diminishing returns to capital. The endogenous growth theory created in the 1980’s is a direct response to the Solow Model as provided dissatisfying accounts of exogenous factors determining long run growth. The fundamentals of the theory were based on the works of Arrow (1962), Romer (1986) and Lucas (1988) but other economists such as Kaldor (1966) and Robinson (1956) have also stressed on technical progress being both a cause and result of economic growth. Arestis sums up the endogenous nature of technical progress well, ‘Technical progress does not fall like manna from heaven, as in neoclassical analysis, but it is the result of deliberate, purposive action induced by economic events’ (Arestis, 1992, pg. 213)

As a matter of fact, it seems rather unrealistic that investment in human capital, innovation and research and development can be completely ignored as significant contributors to economic growth. Countries such as South Korea and Japan have benefited tremendously from investment R&D programmes as well as new and innovative manufacturing processes, especially within the field of electronics and automobiles. Japan has the largest global electronics goods industry and is the world’s third largest automobile manufacturing country. It is also ranked amongst the most innovative countries in the world leaving several measures of global patent filing contributing to its national economy being the third largest in the world. Further, the endogenous growth theory’s approach of imperfect competition and increasing returns to scale, a contrast to the Solow Model, promoted by Romer (1986) and Lucas (1988), was eventually agreed on by Solow himself in 1994: ‘It is perfectly possible to have increasing returns to scale and preserve all the standard neoclassical results’ (Solow, 1994, pg. 49).

Another criticism of the Solow Model is its disregard for the impact of entrepreneurship in generating economic growth. Technical innovation may be exogenously determined but it is still considered to result in economic growth, however, entrepreneurship is abstracted. Both Kaldor as well as Schumpeter have focused on a ‘stream of innovations’ attributed to entrepreneurship (Harcourt, 2006, pg. 134). However, Solow defends his work in when he states, ‘greater allocation of resources to R&D buys a one-time jump in productivity, but not a faster rate of productivity growth’ (Solow, 1994, pg. 53).

A third point for dispute and consequently a limitation of the Solow Model is its assumption that population growth proves to be a hindrance for economic growth, that is, population growth should be controlled to maximise productivity; ceteris paribus, the model proposes that population growth inhibits a country’s ability to develop. Recent studies however, discount this assumption proposing such an effect of population growth to be an exaggerated phenomenon, stating, ‘Population growth does not seem to have the large negative effects that are frequently conjectured’ (Fine, 2000, pg. 51). Contrastingly, many BRIC countries have benefited from their sizeable populations as these have contributed to a large work force and in the case of China, an abundant availability of cheap labour, allowing Chinese goods to be competitive in international markets. Additionally, if a country is experiencing increases in population, it can benefit from it economically if the government is providing education and healthcare to its citizens so they can eventually contribute to making the work force more skilled.

A significant feature of the Solow model is the idea of convergence, i.e. by result of capital accumulation; developing countries will catch up with the developed world and eventually they will be close to the equilibrium capital-output ratio. Historically and empirically today, one does not notice such a trend of convergence. It cannot be denied that many European economies do comply with this notion of convergence (Fig.1), however there is vast disparity between developed and developing economies, contradicting the models analysis (Fig.2): ‘Poor countries are not catching up with the rich’ (Fine, 2000, pg. 51).

An alternative noticeable criticism directed at the Solow Model is its supply-orientated nature, which arguably ‘takes us back to the pre-Keynesian world where demand does not matter for an understanding of the determination of the level of output.’ (Thirlwall, 2002, pg. 25). The models substitution of demand with supply is thus considered backward, as demand is a key determinant to understand growth as a whole. Kaldor, an advocate of the endogenous growth theory, underlines the complimentary nature of demand and supply. Other less prominent but still important criticisms of the model include its failure to acknowledge the importance of institutions and the role of the government in ensuring economic prosperity. Also the models assumption of both labour and capital being homogenous is unrealistic.

Conclusion:

Thus far, this essay has discussed the features and assumptions of the Solow Model as well as its limitations in predicting long-term growth. One has realised that many of the assumptions of the model have been criticised and proposed as its limitations, however, one must understand that every theory must have assumptions: ‘All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorising is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive’ (Solow, 1956, pg. 65). Despite the limitations posed by the model, it cannot be denied that it has had some success in determining long-term economic growth, especially in regard to East Asia. The models explanation of increasing investment rates causing higher output technology is applicable to South Korea, Singapore and Taiwan who have experienced phenomenal economic growth due to increased investment since 1950. Moreover, within the school of economic development, the theory has been a forerunner, achieving widespread acknowledgement and relative success. It has had many setbacks and has been critiqued for its shortcomings but the Solow Model has encouraged much debate about economic growth, investment and capital accumulation, thus making it very influential (Thirlwall, 2003, pg. 144).

Bibliography:

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[ 1 ]. The production function describes how inputs combine together to produce the desired output. In this example the production function has the Cobb-Douglas form. [ 2 ]. ‘g’ is an exogenous positive parameter and can be ignored in this example. [ 3 ]. The y-axis measures output per worker.

[ 4 ]. World Intellectual Property Indicators, 2012 edition, stable URL: http://www.wipo.int/ipstats/en/statistics/patents/