Solow–Swan model
The Solow model ( Solow - Swan model also ) is a technology developed in 1956 by Robert Merton Solow and Trevor Swan mathematical description ( economic model ) the growth of an economy. It forms the basis of neoclassical growth theory.
The Solow model explains growth as a process of accumulation of capital towards a long-term equilibrium between investment and depreciation, the " steady state ". An economy that has little capital to start is in the model save up additional capital and thereby grow - initially high, with increasing capital accumulation, then the low growth - until the long-term equilibrium is reached. In the long-run equilibrium is the growth rate of per capita production is zero. Further growth is only possible through not in the model explained technological progress.
- 2.1 Growth Accounting
- 2.2 convergence
- 2.3 International and historical differences in income
The model
Assumptions
The economy is considered in the Solow model as an aggregate unit (so to speak as a single household), which performs any production and consumption activity. The economy has at any time t certain amounts of capital (K (t)), labor ( L (t) ) and technology (T ( t)), from which together according to a production function F is an output Y is produced:
The product is referred to as "effective work". For the production function F is assumed to be " neoclassical ", ie has the following three properties:
In its simplest form, the Solow model also relates to a closed economy without government activity. Income and production must meet in such an economy, production can therefore be issued either for consumption or for investment:
The investment also correspond exactly in balance with what the economy saves. In a closed economy is thus. The savings behavior of the economy is modeled by a constant savings rate s :, where s is between 0 and 1. Thus, the economy saving in each period a certain percentage of the total production. This is constant over time savings rate s is not accepted as a model in certain exogenous parameters.
Two further assumptions concerning capital and labor: Regarding capital is assumed that in each period a certain percentage of existing capital is unusable ( depreciation) during the working population n at a constant growth rate increases exponentially.
The growth process
For the analysis of economies with a growing population and to improve comparability of economies of different sizes, the model sizes are not absolute, but expressed per head, lower case for per capita quantities are used. It thus defines:
Where the last equation follows from the assumption of constant returns to scale.
Under the assumption of a constant technology can then use the per capita capital, per capita production function be defined as
This gives for each per capita capital stock of k, how much output is produced per capita. For the development of per capita income that is, the per capita capital stock is critical.
Its development is determined by three factors:
Thus, the change in the per capita capital stock each period is given as
If is positive, the per capita capital stock and, thus per capita income grows. If is negative, shrink per capita capital and production. In the long-run equilibrium - the " steady state " level of the economy - will be that the investment exactly the depreciation (taking into account population growth ) correspond to the capital model:
The per capita capital stock k that satisfies this equation is the steady-state capital stock in the economy. The above assumptions on the production function ( constant returns to scale, positive, decreasing marginal profits and the Inada conditions) guarantee the existence of a unique steady-state equilibrium.
Graphically this can be shown on the vertical axis in a chart with per capita capital on the horizontal and per capita income: f ( k) is a concave function according to the assumptions, as well as the economic saving function sf (k). is a straight line, which indicates how much must be saved in order to keep the per capita capital stock just constant and is therefore also referred to as investment demand line or requirement line. The intersection between saving function and investment demand line determines the long- run equilibrium ( steady state ) of the capital stock, which just so much that is saved is the capital stock remains constant despite depreciation and population growth. If this capital stock is reached, the growth rate is zero and per capita production, income and capital are constant over time.
If the per capita capital is below the long-term equilibrium level, the economy will grow and finally reach the long-term equilibrium asymptotically. The growth rate to the number with an increasing capital stock further and further behind - an implication of the assumption that the marginal product of capital decrease. The Solow model predicts, therefore, that, ceteris paribus, economies with lower per capita capital stock to grow faster than those with high capitalization.
Changes in exogenous parameters
The long-term steady-state capital stock is, as stated above, by
The savings rate is s, the depreciation rate and the population growth rate n as exogenous, is not considered certain parameters in the model. However, changes in these parameters affect the long-term equilibrium of the economy.
Population growth and depreciation
A faster population growth ( larger n ) or larger depreciation ( larger δ ) have in the model the same effects on the long-term equilibrium level, you increase the slope of the investment demand curve, thus lowering per capita capital stock and income: must in each period more workers with capital provided (or more capital must be replaced ) be such that for the same savings behavior and the same production technology less per capita capital is formed. Figure 2 shows graphically how the long-term equilibrium responds to an increased population growth: the green line saving function remains unchanged, the investment demand line with slope rotated from the original black in the blue line. The new long-run equilibrium B results from the intersection of the changes in investment demand line with the saving function and is characterized by a lower per capita capital and income than the previous equilibrium A. Since the new investment demand line in the capital stock is higher than is saved up enough capital - the economy is shrinking. This process continues until the new equilibrium level is reached at point B and asymptotically.
Savings rate and golden rule of accumulation
An increase in the saving rate shifts the saving curve of the economy upward, which means that the steady-state per capita capital stock increases and thus the per - capita income. Figure 3 illustrates this graphically: The increase in the savings rate shifts the saving function from their original location (green) to the top ( blue), while investment demand line (black ) remains unchanged. The new equilibrium B results from the intersection of the investment requirement line with the new save feature and has a greater per capita capital stock and higher incomes.
However, the effect of such an increase on consumption is ambiguous: on the one hand, in the long-term equilibrium of production, on the other hand, is associated with higher savings rate but the per capita consumption lower. Graphically equivalent to the per capita consumption for a given per capita capital to the vertical distance between the production function and the saving function on the same per capita capital; In Figure 2, this is the first vertical distance between the red and the green line, and later that between the red and the blue line. This shows why the effect is fundamentally indeterminate on consumption: Although, the new equilibrium point B in a higher production per capita, the new saving function, however, is also closer to the production function.
The golden rule of accumulation describes that savings rate in an economy, the consumer is maximized. Apply to each savings rate s must be in the long-run equilibrium that. At the same time belonging to the equilibrium level of consumption is given by. For this reason, the equilibrium consumption can be described as a function of the saving rate:
This can then be maximized over s and yields the first-order condition:
Wherein, so that the condition can be simplified to. The capital stock that satisfies this equation, and the associated savings rate to maximize the long-term equilibrium consumption in the economy. Although the savings rate found in this way maximizes the equilibrium consumption, it is not clear whether this is desirable from the perspective of an economy. For an economy that is in equilibrium with, means an increase in the savings ratio, although a persistently higher level of consumption, but this is only when the new equilibrium is reached. In the first period after the increase in the savings rate, consumption, however, would first reduce (since the savings rate is increased, but not enough capital was formed for the new steady state, and therefore the production in comparison to the new steady state is still low ). Depending on how much weight puts the economy on today 's future over consumption, so it might not be desirable to increase the savings rate today in order to achieve in the long run for a new equilibrium with higher consumption. The situation is different if the current savings rate. In this case, an equilibrium with higher consumption could be achieved by the savings rate reduced, would therefore consumes more. The economy would thereby consume more and more in the new equilibrium and also in the periods before that. A situation is therefore referred to as " dynamically inefficient ".
Figures 4 and 5 show graphically how can the effects of different changes in the savings rate. In Figure 4, the savings rate increased from an original, dynamic, efficient equilibrium. The increase leads to positive capital growth and increasing capital and income per capita. The growth of capital decreases with increasing capital accumulation and goes asymptotically to zero, the economy reaches a new equilibrium with a higher capital income and consumption per capita. However, at the beginning of the process of this long-term higher level needs to be "bought " with lower consumption. Whether such a change in the savings rate from the perspective of the national economy is desirable, therefore depends on how early consumption loss compared to the later consumption gain is evaluated. Figure 5 shows the savings rate is lowered, starting from a dynamically inefficient, ie high savings rate. The reduction leads to a negative capital growth and thus decreasing capital and income per capita. The capital reduction decreases with increasing capital accumulation and goes asymptotically to zero, the economy reaches a new equilibrium with lower capital and income per capita. The long-term consumption per capita is higher, there is less conserved. The central difference for dynamically efficient situation is that the consumer is not only long-term higher, but in each period from the increase. The economy can not only consume in the long term by lowering the savings rate more, but immediately. Regardless of the evaluation early to later use is desirable in a savings rate above the "Golden " savings rate, a reduction in the savings rate from the perspective of the national economy so definitely.
Figure 5: reduction in the savings rate from a dynamically inefficient situation. On the respective vertical axis the indicated sizes are shown, the horizontal axis indicates the time. The change in the savings rate takes place in period 100
Technological advancement
Technological progress shifts the production function and thus the saving function in ky diagram upwards; the new intersection with the investment requirement line is thus at a higher per capita capital and income level. Technological progress can therefore lead thus also in the long-term equilibrium growth.
With technological progress, the production factor labor multiplied, and under the assumptions described in Section 1.1, the production function can be divided by the factor. This is instead of the per capita production previously used output per effective labor unit, depending on the stock of capital per effective unit of work:
The need for investment per effective unit of work is as far from the depreciation rate and the growth of the population; now but also the greater as a result of technological progress, labor productivity must also be balanced: Technological progress leads to an increase in effective labor units (the product increases), the capital per effective unit of work thus decreases ceteris paribus. Assuming an exponential technology growth with growth rate, the investment demand curve for capital per effective unit of work as a result. The equation of motion for capital per effective unit of work is thus:
The long-term equilibrium is reached, if the capital stock per effective labor unit is constant, ie, if
The capital stock and thus the income per effective labor unit does not grow so the long-term equilibrium. The per capita income, however, is given by. It grows at the same rate as the technology of the economy. A growth of per capita variables in the long-run equilibrium is possible, but only because of exogenous technological progress.
Example: Solow model with Cobb -Douglas production function
A possible production function that satisfies the assumptions described above, the Cobb - Douglas function :, where is. After dividing by the population size contributes to the per capita version. The equation of motion of per capita capital stocks is then given by. The long-run equilibrium is obtained when this change is zero and hence
And so then
Since a greater savings rate and a higher level of technology, however, lead to a higher equilibrium capital stock, faster population growth and a greater rate of depreciation at a lower balance.
Empirical applications
Growth Accounting
Closely associated with the Solow model is the so-called " growth accounting " ( growth accounting ), which was driven by Moses Abramovitz and Robert Solow. It is investigated how much of the economic growth can be explained by capital, labor and other factors. For a general function of the form of production can be shown that the growth of the total production may be divided by means of
The elasticity of production with regard to state on capital. In this way, the per capita economic growth can be divided into per capita growth due to per capita capital accumulation and a further term R (t ), the so-called Solow residual. This is sometimes interpreted as the contribution of technological progress to growth, but is actually a collective term for all the factors that lead to economic growth and are not already covered by the accumulation of capital.
Convergence
If the economy is still below the long -term equilibrium level and growing, its growth rate is higher, the lower the per capita capital stock. An economy with initially low per capita capital, according to the Solow model thus initially have very high growth rates, which then decrease with increasing capital accumulation and eventually tend to 0. For two economies with the same technology and the same exogenous parameters ( depreciation rate, savings rate, population growth ) and thus the same long-run equilibrium, but different initial capital equipment, the model predicts that the initially poorer economies grow faster and thus " catch up " with respect to the originally richer economy will. This process is referred to as " convergence ". However, the Solow model predicts no " absolute convergence " before, in the catch all poor countries alike, converge to the same long-run equilibrium; The hypothesis of the Solow model is instead the " conditional convergence " whereby countries grow faster the further they are away from their specific long-term equilibrium. A prediction of the model would thus not be that poor countries grow faster than rich, but that have the originally poorer higher growth rates among "similar" countries. In fact, among the OECD countries, a negative correlation between their per capita income in 1960 and the average annual growth rate between 1960 and 2000. For even more significant negative correlation exists between the per capita income of the states in the United States in 1880 and their annual growth rates 1880-2000.
Another test for convergence was conducted by N. Gregory Mankiw, David Romer and David N. Weil 1992. Based on a sample of 98 countries, they showed that there was no correlation between the per capita income in 1960 and the growth rate 1960-1985, so no absolute convergence. However, if the investment rate and population growth is statistically controlled, the negative effect of the initial income levels, which supports the hypothesis of conditional convergence shows. The standard Solow model overestimated here, however, the speed of convergence, the actual catch-up is slower than predicted by the model. Mankiw, Romer and Weil, however able to show that an order of human capital as a third factor of production extended Solow model predicts that in about convergence speed, which is also visible in the data. In this extended model, the production function and factor H from human capital and is
With. Human capital per effective unit of work is similar, which according to an equation of motion that for capital per effective unit of work: , where the ( also exogenous ) refers to investment ratio for human capital. At steady state, then capital and human capital per effective labor unit constant.
International and historical differences in income
According to the Solow model, there are two possible reasons for per capita income differentials between economies: a different per capita capital stock or different labor productivity. In fact, however, the per capita capital stock can not explain the large differences in income between rich and poor countries today, or between developed countries past and present. The per capita income of industrialized countries is around ten times larger than a century earlier; the natural logarithms of per capita income today and 100 years ago so different about him. With the definition of the elasticity of per capita income in terms of per capita capital, follows. With follows that the per capita capital stock difference
Must be. Empirical studies suggest that around one third. If the per capita capital would be the only source for per capita income differences, the per capita capital so would have to be grown by about a factor of 1000 in the last few years, a growth in per capita income by a factor of 10 to explain. In fact, the per capita capital is grown only by about a factor of 10. The growth of per capita capital can not explain the extent of economic growth over the last 100 years so.
Genesis
Solow and Swan independently developed similar versions of their growth model; Solow published his post in February 1956 in the Quarterly Journal of Economics, Swans article appeared in November in Economic Record. While bumped at the start and Swans Product to large professional reception - the post has been taken several times in anthologies, and Swan was invited as a visiting professor at various universities - sat down in the long term by Solow version of the model and in particular its choice of graphical representation.
The Solow - Swan model was a critique and further development of prevailing at that time growth model according to Harrod - Domar. As the Solow model also took the Harrod - Domar model to a constant, exogenous savings rate. The model was based also on a constant marginal productivity of capital and on a production function with low or non-existent substitutability between labor and capital. The Harrod - Domar model allows several different Steady States: In one possible scenario, capital grows without being used in another growing unemployment. Only in a parameter constellation results in a steady state, in which all the available factors of production are used.
Criticism and further developments
The basic Solow model assumes a closed economy without government activity. An involvement of the public sector and international capital flows is, however, possible.
However, a central assumption of the Solow model is the exogenously given, over time constant savings rate. This means that an economy independent of the level of their income always saves the same the same percentage. The savings behavior is therefore not modeled, and therefore can not be investigated how the economy responds to changes in the interest rate or the capital tax rate. In addition, empirical studies also suggest that the savings rate increases with increasing income. An important extension of the Solow model is therefore to formulate the savings rate as a function of income, which requires an explicit modeling of the saving behavior of households. Such was introduced by Ramsey in 1928 and then further developed by Cass (1965) and Koopmans (1965). The resulting model is therefore often referred to as Ramsey or Ramsey -Cass - Koopmans model.
The Solow model does not explain further what is meant by "technology" or " labor productivity ". It is a collective term for all factors that affect the per - capita income and are not already included in capital and labor. For this purpose, among others, could the education of the working population, the infrastructure, but also political institutions such as property rights belong. Furthermore, the model takes this central for the model Wachstumsdeterminante as exogenously given. While the Ramsey -Cass - Koopmans model abolished the Endogenisieriung the savings rate, it retained the assumption of exogenous technological progress. Criticism of this assumption led in the late 1980s to the development of so-called endogenous growth models ( endogenous growth models ), which, among other important Paul Romer, Philippe Aghion and Peter Howitt and Gene M. Grossman and Elhanan Helpman posts written (see also Romer model or Jones model). In these models, technological progress is not seen as externally preset size, but is determined endogenously within the model.