Solow–Swan model

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The Solow–Swan model is an economic model of long-run economic growth set within the framework of neoclassical economics. It attempts to explain long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity, commonly referred to as technological progress. At its core is a neoclassical (aggregate) production function, often specified to be of Cobb–Douglas type, which enables the model "to make contact with microeconomics". [1] :26 The model was developed independently by Robert Solow and Trevor Swan in 1956, [2] [3] [note 1] and superseded the Keynesian Harrod–Domar model.

Economic model simplified representations of economic reality

In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework designed to illustrate complex processes. Frequently, economic models posit structural parameters. A model may have various exogenous variables, and those variables may change to create various responses by economic variables. Methodological uses of models include investigation, theorizing, and fitting theories to the world.

Economic growth increase in production and consumption in an economy

Economic growth is the increase in the inflation-adjusted market value of the goods and services produced by an economy over time. It is conventionally measured as the percent rate of increase in real gross domestic product, or real GDP.

Neoclassical economics is an approach to economics focusing on the determination of goods, outputs, and income distributions in markets through supply and demand. This determination is often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits by firms facing production costs and employing available information and factors of production, in accordance with rational choice theory, a theory that has come under considerable question in recent years.


Mathematically, the Solow–Swan model is a nonlinear system consisting of a single ordinary differential equation that models the evolution of the per capita stock of capital. Due to its particularly attractive mathematical characteristics, Solow–Swan proved to be a convenient starting point for various extensions. For instance, in 1965, David Cass and Tjalling Koopmans integrated Frank Ramsey's analysis of consumer optimization, thereby endogenizing the saving rate, to create what is now known as the Ramsey–Cass–Koopmans model.

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

Per capita is a Latin prepositional phrase: per and capita. The phrase thus means "by heads" or "for each head", i.e., per individual/person. The term is used in a wide variety of social sciences and statistical research contexts, including government statistics, economic indicators, and built environment studies.


The neo-classical model was an extension to the 1946 Harrod–Domar model that included a new term: productivity growth. Important contributions to the model came from the work done by Solow and by Swan in 1956, who independently developed relatively simple growth models. [2] [3] Solow's model fitted available data on US economic growth with some success. [4] In 1987 Solow was awarded the Nobel Prize in Economics for his work. Today, economists use Solow's sources-of-growth accounting to estimate the separate effects on economic growth of technological change, capital, and labor. [5]

United States Federal republic in North America

The United States of America (USA), commonly known as the United States or simply America, is a country comprising 50 states, a federal district, five major self-governing territories, and various possessions. At 3.8 million square miles, the United States is the world's third or fourth largest country by total area and is slightly smaller than the entire continent of Europe. Most of the country is located in central North America between Canada and Mexico. With an estimated population of over 327 million people, the U.S. is the third most populous country. The capital is Washington, D.C., and the most populous city is New York City.

Extension to the Harrod–Domar model

Solow extended the Harrod–Domar model by adding labor as a factor of production and capital-output ratios that are not fixed as they are in the Harrod–Domar model. These refinements allow increasing capital intensity to be distinguished from technological progress. Solow sees the fixed proportions production function as a "crucial assumption" to the instability results in the Harrod-Domar model. His own work expands upon this by exploring the implications of alternative specifications, namely the Cobb–Douglas and the more general constant elasticity of substitution (CES). [2] Although this has become the canonical and celebrated story [6] in the history of economics, featured in many economic textbooks, [7] recent reappraisal of Harrod's work has contested it. One central criticism is that Harrod's original piece [8] was neither mainly concerned with economic growth nor did he explicitly use a fixed proportions production function. [7] [9]

Capital intensity is the amount of fixed or real capital present in relation to other factors of production, especially labor. At the level of either a production process or the aggregate economy, it may be estimated by the capital to labor ratio, such as from the points along a capital/labor isoquant.

Leontief production function

In economics, the Leontief production function or fixed proportions production function is a production function that implies the factors of production will be used in fixed proportions, as there is no substitutability between factors. It was named after Wassily Leontief and represents a limiting case of the constant elasticity of substitution production function.

Cobb–Douglas production function

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.

Long-run implications

A standard Solow model predicts that in the long run, economies converge to their steady state equilibrium and that permanent growth is achievable only through technological progress. Both shifts in saving and in populational growth cause only level effects in the long-run (i.e. in the absolute value of real income per capita). An interesting implication of Solow's model is that poor countries should grow faster and eventually catch-up to richer countries. This convergence could be explained by: [10]

Steady-state economy economy made up of constant physical wealth and population size

A steady-state economy is an economy made up of a constant stock of physical wealth (capital) and a constant population size. In effect, such an economy does not grow in the course of time. The term usually refers to the national economy of a particular country, but it is also applicable to the economic system of a city, a region, or the entire world. Early in the history of economic thought, classical economist Adam Smith of the 18th century developed the concept of a stationary state of an economy: Smith believed that any national economy in the world would sooner or later settle in a final state of stationarity.

The idea of convergence in economics is the hypothesis that poorer economies' per capita incomes will tend to grow at faster rates than richer economies. As a result, all economies should eventually converge in terms of per capita income. Developing countries have the potential to grow at a faster rate than developed countries because diminishing returns are not as strong as in capital-rich countries. Furthermore, poorer countries can replicate the production methods, technologies, and institutions of developed countries.

In economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker.

Baumol attempted to verify this empirically and found a very strong correlation between a countries' output growth over a long period of time (1870 to 1979) and its initial wealth. [11] His findings were later contested by DeLong who claimed that both the non-randomness of the sampled countries, and potential for significant measurement errors for estimates of real income per capita in 1870, biased Baumol's findings. DeLong concludes that there is little evidence to support the convergence theory.


The key assumption of the neoclassical growth model is that capital is subject to diminishing returns in a closed economy.

Variations in the effects of productivity

In the Solow–Swan model the unexplained change in the growth of output after accounting for the effect of capital accumulation is called the Solow residual. This residual measures the exogenous increase in total factor productivity (TFP) during a particular time period. The increase in TFP is often attributed entirely to technological progress, but it also includes any permanent improvement in the efficiency with which factors of production are combined over time. Implicitly TFP growth includes any permanent productivity improvements that result from improved management practices in the private or public sectors of the economy. Paradoxically, even though TFP growth is exogenous in the model, it cannot be observed, so it can only be estimated in conjunction with the simultaneous estimate of the effect of capital accumulation on growth during a particular time period.

The model can be reformulated in slightly different ways using different productivity assumptions, or different measurement metrics:

In a growing economy, capital is accumulated faster than people are born, so the denominator in the growth function under the MFP calculation is growing faster than in the ALP calculation. Hence, MFP growth is almost always lower than ALP growth. (Therefore, measuring in ALP terms increases the apparent capital deepening effect.) MFP is measured by the "Solow residual", not ALP.

Mathematics of the model

The textbook Solow–Swan model is set in continuous-time world with no government or international trade. A single good (output) is produced using two factors of production, labor () and capital () in an aggregate production function that satisfies the Inada conditions, which imply that the elasticity of substitution must be asymptotically equal to one. [12] [13]

where denotes time, is the elasticity of output with respect to capital, and represents total production. refers to labor-augmenting technology or “knowledge”, thus represents effective labor. All factors of production are fully employed, and initial values , , and are given. The number of workers, i.e. labor, as well as the level of technology grow exogenously at rates and , respectively:

The number of effective units of labor, , therefore grows at rate . Meanwhile, the stock of capital depreciates over time at a constant rate . However, only a fraction of the output ( with ) is consumed, leaving a saved share for investment:

where is shorthand for , the derivative with respect to time. Derivative with respect to time means that it is the change in capital stock—output that is neither consumed nor used to replace worn-out old capital goods is net investment.

Since the production function has constant returns to scale, it can be written as output per effective unit of labour: [note 2]

The main interest of the model is the dynamics of capital intensity , the capital stock per unit of effective labour. Its behaviour over time is given by the key equation of the Solow–Swan model: [note 3]

The first term, , is the actual investment per unit of effective labour: the fraction of the output per unit of effective labour that is saved and invested. The second term, , is the “break-even investment”: the amount of investment that must be invested to prevent from falling. [14] :16 The equation implies that converges to a steady-state value of , defined by , at which there is neither an increase nor a decrease of capital intensity:

at which the stock of capital and effective labour are growing at rate . By assumption of constant returns, output is also growing at that rate. In essence, the Solow–Swan model predicts that an economy will converge to a balanced-growth equilibrium, regardless of its starting point. In this situation, the growth of output per worker is determined solely by the rate of technological progress. [14] :18

Since, by definition, , at the equilibrium we have

Therefore, at the equilibrium, the capital/output ratio depends only on the saving, growth, and depreciation rates. This is the Solow–Swan model's version of the golden rule saving rate.

Since , at any time the marginal product of capital in the Solow–Swan model is inversely related to the capital/labor ratio.

If productivity is the same across countries, then countries with less capital per worker have a higher marginal product, which would provide a higher return on capital investment. As a consequence, the model predicts that in a world of open market economies and global financial capital, investment will flow from rich countries to poor countries, until capital/worker and income/worker equalize across countries.

Since the marginal product of physical capital is not higher in poor countries than in rich countries, [15] the implication is that productivity is lower in poor countries. The basic Solow model cannot explain why productivity is lower in these countries. Lucas suggested that lower levels of human capital in poor countries could explain the lower productivity. [16]

If one equates the marginal product of capital with the rate of return (such approximation is often used in neoclassical economics), then, for our choice of the production function

so that is the fraction of income appropriated by capital. Thus, Solow–Swan model assumes from the beginning that the labor-capital split of income remains constant.

Mankiw–Romer–Weil version of model

Addition of human capital

N. Gregory Mankiw, David Romer, and David Weil created a human capital augmented version of the Solow–Swan model that can explain the failure of international investment to flow to poor countries. [17] In this model output and the marginal product of capital (K) are lower in poor countries because they have less human capital than rich countries.

Similar to the textbook Solow–Swan model, the production function is of Cobb–Douglas type:

where is the stock of human capital, which depreciates at the same rate as physical capital. For simplicity, they assume the same function of accumulation for both types of capital. Like in Solow–Swan, a fraction of the outcome, , is saved each period, but in this case split up and invested partly in physical and partly in human capital, such that . Therefore, there are two fundamental dynamic equations in this model:

The balanced (or steady-state) equilibrium growth path is determined by , which means and . Solving for the steady-state level of and yields:

In the steady state, .

Econometric estimates

Klenow and Rodriguez-Clare cast doubt on the validity of the augmented model because Mankiw, Romer, and Weil's estimates of did not seem consistent with accepted estimates of the effect of increases in schooling on workers' salaries. Though the estimated model explained 78% of variation in income across countries, the estimates of implied that human capital's external effects on national income are greater than its direct effect on workers' salaries. [18]

Accounting for external effects

Theodore Breton provided an insight that reconciled the large effect of human capital from schooling in the Mankiw, Romer and Weil model with the smaller effect of schooling on workers' salaries. He demonstrated that the mathematical properties of the model include significant external effects between the factors of production, because human capital and physical capital are multiplicative factors of production. [19] The external effect of human capital on the productivity of physical capital is evident in the marginal product of physical capital:

He showed that the large estimates of the effect of human capital in cross-country estimates of the model are consistent with the smaller effect typically found on workers' salaries when the external effects of human capital on physical capital and labor are taken into account. This insight significantly strengthens the case for the Mankiw, Romer, and Weil version of the Solow–Swan model. Most analyses criticizing this model fail to account for the pecuniary external effects of both types of capital inherent in the model. [19]

Total factor productivity

The exogenous rate of TFP (total factor productivity) growth in the Solow–Swan model is the residual after accounting for capital accumulation. The Mankiw, Romer, and Weil model provide a lower estimate of the TFP (residual) than the basic Solow–Swan model because the addition of human capital to the model enables capital accumulation to explain more of the variation in income across countries. In the basic model, the TFP residual includes the effect of human capital because human capital is not included as a factor of production.

Conditional convergence

The Solow–Swan model augmented with human capital predicts that the income levels of poor countries will tend to catch up with or converge towards the income levels of rich countries if the poor countries have similar savings rates for both physical capital and human capital as a share of output, a process known as conditional convergence. However, savings rates vary widely across countries. In particular, since considerable financing constraints exist for investment in schooling, savings rates for human capital are likely to vary as a function of cultural and ideological characteristics in each country. [20]

Since the 1950s, output/worker in rich and poor countries generally has not converged, but those poor countries that have greatly raised their savings rates have experienced the income convergence predicted by the Solow–Swan model. As an example, output/worker in Japan, a country which was once relatively poor, has converged to the level of the rich countries. Japan experienced high growth rates after it raised its savings rates in the 1950s and 1960s, and it has experienced slowing growth of output/worker since its savings rates stabilized around 1970, as predicted by the model.

The per-capita income levels of the southern states of the United States have tended to converge to the levels in the Northern states. The observed convergence in these states is also consistent with the conditional convergence concept. Whether absolute convergence between countries or regions occurs depends on whether they have similar characteristics, such as:

Additional evidence for conditional convergence comes from multivariate, cross-country regressions. [22]

Econometric analysis on Singapore and the other "East Asian Tigers" has produced the surprising result that although output per worker has been rising, almost none of their rapid growth had been due to rising per-capita productivity (they have a low "Solow residual"). [5]

See also


  1. The idea of using a Cobb–Douglas production function at the core of a growth model dates back to Tinbergen, J. (1942). "Zur Theorie der langfristigen Wirtschaftsentwicklung". Weltwirtschaftliches Archiv . 55: 511–549. JSTOR   40430851 . See Brems, Hans (1986). "Neoclassical Growth: Tinbergen and Solow". Pioneering Economic Theory, 1630–1980. Baltimore: Johns Hopkins University Press. pp. 362–368. ISBN   978-0-8018-2667-2.
  2. Step-by-step calculation:
  3. Step-by-step calculation: . Since , and , are and , respectively, the equation simplifies to . As mentioned above, .

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Further reading