# Solow–Swan model

Last updated

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 [1] :26 The model was developed independently by Robert Solow and Trevor Swan in 1956, [2] [3] [note 1] and superseded the Keynesian Harrod–Domar model.

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 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.

## Contents

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.

## Background

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]

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.

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.

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]

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.

• Lags in the diffusion on knowledge. Differences in real income might shrink as poor countries receive better technology and information;
• Efficient allocation of international capital flows, since the rate of return on capital should be higher in poorer countries. In practice, this is seldom observed and is known as Lucas' paradox;
• A mathematical implication of the model (assuming poor countries have not yet reached their steady state).

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.

### Assumptions

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:

• Average Labor Productivity (ALP) is economic output per labor hour.
• Multifactor productivity (MFP) is output divided by a weighted average of capital and labor inputs. The weights used are usually based on the aggregate input shares either factor earns. This ratio is often quoted as: 33% return to capital and 67% return to labor (in Western nations).

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 (${\displaystyle L}$) and capital (${\displaystyle K}$) 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]

${\displaystyle Y(t)=K(t)^{\alpha }(A(t)L(t))^{1-\alpha }\,}$

where ${\displaystyle t}$ denotes time, ${\displaystyle 0<\alpha <1}$ is the elasticity of output with respect to capital, and ${\displaystyle Y(t)}$ represents total production. ${\displaystyle A}$ refers to labor-augmenting technology or “knowledge”, thus ${\displaystyle AL}$ represents effective labor. All factors of production are fully employed, and initial values ${\displaystyle A(0)}$, ${\displaystyle K(0)}$, and ${\displaystyle L(0)}$ are given. The number of workers, i.e. labor, as well as the level of technology grow exogenously at rates ${\displaystyle n}$ and ${\displaystyle g}$, respectively:

${\displaystyle L(t)=L(0)e^{nt}}$
${\displaystyle A(t)=A(0)e^{gt}}$

The number of effective units of labor, ${\displaystyle A(t)L(t)}$, therefore grows at rate ${\displaystyle (n+g)}$. Meanwhile, the stock of capital depreciates over time at a constant rate ${\displaystyle \delta }$. However, only a fraction of the output (${\displaystyle cY(t)}$ with ${\displaystyle 0) is consumed, leaving a saved share ${\displaystyle s=1-c}$ for investment:

${\displaystyle {\dot {K}}(t)=s\cdot Y(t)-\delta \cdot K(t)\,}$

where ${\displaystyle {\dot {K}}}$ is shorthand for ${\displaystyle {\frac {dK(t)}{dt}}}$, 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 ${\displaystyle Y(K,AL)}$ has constant returns to scale, it can be written as output per effective unit of labour: [note 2]

${\displaystyle y(t)={\frac {Y(t)}{A(t)L(t)}}=k(t)^{\alpha }}$

The main interest of the model is the dynamics of capital intensity ${\displaystyle k}$, 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]

${\displaystyle {\dot {k}}(t)=sk(t)^{\alpha }-(n+g+\delta )k(t)}$

The first term, ${\displaystyle sk(t)^{\alpha }=sy(t)}$, is the actual investment per unit of effective labour: the fraction ${\displaystyle s}$ of the output per unit of effective labour ${\displaystyle y(t)}$ that is saved and invested. The second term, ${\displaystyle (n+g+\delta )k(t)}$, is the “break-even investment”: the amount of investment that must be invested to prevent ${\displaystyle k}$ from falling. [14] :16 The equation implies that ${\displaystyle k(t)}$ converges to a steady-state value of ${\displaystyle k^{*}}$, defined by ${\displaystyle sk(t)^{\alpha }=(n+g+\delta )k(t)}$, at which there is neither an increase nor a decrease of capital intensity:

${\displaystyle k^{*}=\left({\frac {s}{n+g+\delta }}\right)^{1/(1-\alpha )}\,}$

at which the stock of capital ${\displaystyle K}$ and effective labour ${\displaystyle AL}$ are growing at rate ${\displaystyle (n+g)}$. By assumption of constant returns, output ${\displaystyle Y}$ 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, ${\displaystyle {\frac {K(t)}{Y(t)}}=k(t)^{1-\alpha }}$, at the equilibrium ${\displaystyle k^{*}}$ we have

${\displaystyle {\frac {K(t)}{Y(t)}}={\frac {s}{n+g+\delta }}}$

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 ${\displaystyle {\alpha }<1}$, at any time ${\displaystyle t}$ the marginal product of capital ${\displaystyle K(t)}$ in the Solow–Swan model is inversely related to the capital/labor ratio.

${\displaystyle MPK={\frac {\partial Y}{\partial K}}={\frac {\alpha A^{1-\alpha }}{(K/L)^{1-\alpha }}}}$

If productivity ${\displaystyle A}$ is the same across countries, then countries with less capital per worker ${\displaystyle K/L}$ 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 ${\displaystyle K/L}$ and income/worker ${\displaystyle Y/L}$ 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 ${\displaystyle {\frac {\partial Y}{\partial K}}}$ with the rate of return ${\displaystyle r}$ (such approximation is often used in neoclassical economics), then, for our choice of the production function

${\displaystyle \alpha ={\frac {K{\frac {\partial Y}{\partial K}}}{Y}}={\frac {rK}{Y}}\,}$

so that ${\displaystyle \alpha }$ 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

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:

${\displaystyle Y(t)=K(t)^{\alpha }H(t)^{\beta }(A(t)L(t))^{1-\alpha -\beta },}$

where ${\displaystyle H(t)}$ is the stock of human capital, which depreciates at the same rate ${\displaystyle \delta }$ 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, ${\displaystyle sY(t)}$, is saved each period, but in this case split up and invested partly in physical and partly in human capital, such that ${\displaystyle s=s_{K}+s_{H}}$. Therefore, there are two fundamental dynamic equations in this model:

${\displaystyle {\dot {k}}=s_{K}k^{\alpha }h^{\beta }-(n+g+\delta )k}$
${\displaystyle {\dot {h}}=s_{H}k^{\alpha }h^{\beta }-(n+g+\delta )h}$

The balanced (or steady-state) equilibrium growth path is determined by ${\displaystyle {\dot {k}}={\dot {h}}=0}$, which means ${\displaystyle s_{K}k^{\alpha }h^{\beta }-(n+g+\delta )k=0}$ and ${\displaystyle s_{H}k^{\alpha }h^{\beta }-(n+g+\delta )h=0}$. Solving for the steady-state level of ${\displaystyle k}$ and ${\displaystyle h}$ yields:

${\displaystyle k^{*}=\left({\frac {s_{K}^{1-\beta }s_{H}^{\beta }}{n+g+\delta }}\right)^{\frac {1}{1-\alpha -\beta }}}$
${\displaystyle h^{*}=\left({\frac {s_{K}^{\alpha }s_{H}^{1-\alpha }}{n+g+\delta }}\right)^{\frac {1}{1-\alpha -\beta }}}$

In the steady state, ${\displaystyle y^{*}=(k^{*})^{\alpha }(h^{*})^{\beta }}$.

### Econometric estimates

Klenow and Rodriguez-Clare cast doubt on the validity of the augmented model because Mankiw, Romer, and Weil's estimates of ${\displaystyle {\beta }}$ 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 ${\displaystyle {\beta }}$ 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:

${\displaystyle MPK={\frac {\partial Y}{\partial K}}={\frac {\alpha A^{1-\alpha }(H/L)^{\beta }}{(K/L)^{1-\alpha }}}}$

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]

## Notes

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: ${\displaystyle y(t)={\frac {Y(t)}{A(t)L(t)}}={\frac {K(t)^{\alpha }(A(t)L(t))^{1-\alpha }}{A(t)L(t)}}={\frac {K(t)^{\alpha }}{(A(t)L(t))^{\alpha }}}=k(t)^{\alpha }}$
3. Step-by-step calculation: ${\displaystyle {\dot {k}}(t)={\frac {{\dot {K}}(t)}{A(t)L(t)}}-{\frac {K(t)}{[A(t)L(t)]^{2}}}[A(t){\dot {L}}(t)+L(t){\dot {A}}(t)]={\frac {{\dot {K}}(t)}{A(t)L(t)}}-{\frac {K(t)}{A(t)L(t)}}{\frac {{\dot {L}}(t)}{L(t)}}-{\frac {K(t)}{A(t)L(t)}}{\frac {{\dot {A}}(t)}{A(t)}}}$. Since ${\displaystyle {\dot {K}}(t)=sY(t)-\delta K(t)\,}$, and ${\displaystyle {\frac {{\dot {L}}(t)}{L(t)}}}$, ${\displaystyle {\frac {{\dot {A}}(t)}{A(t)}}}$ are ${\displaystyle n}$ and ${\displaystyle g}$, respectively, the equation simplifies to ${\displaystyle {\dot {k}}(t)=s{\frac {Y(t)}{A(t)L(t)}}-\delta {\frac {K(t)}{A(t)L(t)}}-n{\frac {K(t)}{A(t)L(t)}}-g{\frac {K(t)}{A(t)L(t)}}=sy(t)-\delta k(t)-nk(t)-gk(t)}$. As mentioned above, ${\displaystyle y(t)=k(t)^{\alpha }}$.

## Related Research Articles

In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

Growth accounting is a procedure used in economics to measure the contribution of different factors to economic growth and to indirectly compute the rate of technological progress, measured as a residual, in an economy. Growth accounting decomposes the growth rate of an economy's total output into that which is due to increases in the contributing amount of the factors used—usually the increase in the amount of capital and labor—and that which cannot be accounted for by observable changes in factor utilization. The unexplained part of growth in GDP is then taken to represent increases in productivity or a measure of broadly defined technological progress.

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:

1. With a shape parameter k and a scale parameter θ.
2. With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
3. With a shape parameter k and a mean parameter μ = = α/β.

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

In economics, Okun's law is an empirically observed relationship between unemployment and losses in a country's production. The "gap version" states that for every 1% increase in the unemployment rate, a country's GDP will be roughly an additional 2% lower than its potential GDP. The "difference version" describes the relationship between quarterly changes in unemployment and quarterly changes in real GDP. The stability and usefulness of the law has been disputed.

In economics and in particular neoclassical economics, the marginal product or marginal physical productivity of an input is the change in output resulting from employing one more unit of a particular input, assuming that the quantities of other inputs are kept constant.

James Edward Meade, was a British economist and winner of the 1977 Nobel Memorial Prize in Economic Sciences jointly with the Swedish economist Bertil Ohlin for their "pathbreaking contribution to the theory of international trade and international capital movements."

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:

The Gauss–Newton algorithm is used to solve non-linear least squares problems. It is a modification of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.

The Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow, the Nobel Memorial Prize in Economic Sciences-winning economist, defined rising productivity as rising output with constant capital and labor input. It is a "residual" because it is the part of growth that is not accounted for by measures of capital accumulation or increased labor input. Increased physical throughput – i.e. environmental resources – is specifically excluded from the calculation; thus some portion of the residual can be ascribed to increased physical throughput. The example used is for the intracapital substitution of aluminium fixtures for steel during which the inputs do not alter. This differs in almost every other economic circumstance in which there are many other variables. The Solow Residual is procyclical and measures of it are now called the rate of growth of multifactor productivity or total factor productivity, though Solow (1957) did not use these terms.

The Harrod–Domar model is a classical Keynesian model of economic growth. It is used in development economics to explain an economy's growth rate in terms of the level of saving and productivity of capital. It suggests that there is no natural reason for an economy to have balanced growth. The model was developed independently by Roy F. Harrod in 1939, and Evsey Domar in 1946, although a similar model had been proposed by Gustav Cassel in 1924. The Harrod–Domar model was the precursor to the exogenous growth model.

The Ramsey–Cass–Koopmans model, or Ramsey growth model, is a neoclassical model of economic growth based primarily on the work of Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans. The Ramsey–Cass–Koopmans model differs from the Solow–Swan model in that the choice of consumption is explicitly microfounded at a point in time and so endogenizes the savings rate. As a result, unlike in the Solow–Swan model, the saving rate may not be constant along the transition to the long run steady state. Another implication of the model is that the outcome is Pareto optimal or Pareto efficient.

The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product .

The Goodwin model, sometimes called Goodwin’s class struggle model, is a model of endogenous economic fluctuations first proposed by the American economist Richard M. Goodwin in 1967. It combines aspects of the Harrod–Domar growth model with the Phillips curve to generate endogenous cycles in economic activity unlike most modern macroeconomic models in which movements in economic aggregates are driven by exogenously assumed shocks. Since Goodwin's publication in 1967, the model has been extended and applied in various ways.

The AK model of economic growth is an endogenous growth model used in the theory of economic growth, a subfield of modern macroeconomics. In the 1980s it became progressively clearer that the standard neoclassical exogenous growth models were theoretically unsatisfactory as tools to explore long run growth, as these models predicted economies without technological change and thus they would eventually converge to a steady state, with zero per capita growth. A fundamental reason for this is the diminishing return of capital; the key property of AK endogenous-growth model is the absence of diminishing returns to capital. In lieu of the diminishing returns of capital implied by the usual parameterizations of a Cobb–Douglas production function, the AK model uses a linear model where output is a linear function of capital. Its appearance in most textbooks is to introduce endogenous growth theory.

In the technological theory of social production, the growth of output, measured in money units, is related to achievements in technological consumption of labour and energy. This theory is based on concepts of classical political economy and neo-classical economics and appears to be a generalisation of the known economic models, such as the neo-classical model of economic growth and input-output model.

Uzawa's theorem, also known as the steady state growth theorem, is a theorem in economic growth theory concerning the form that technological change can take in the Solow–Swan and Ramsey–Cass–Koopmans growth models. It was first proved by Japanese economist Hirofumi Uzawa.

## References

1. Acemoglu, Daron (2009). "The Solow Growth Model". Introduction to Modern Economic Growth. Princeton: Princeton University Press. pp. 26–76. ISBN   978-0-691-13292-1.
2. Solow, Robert M. (February 1956). "A contribution to the theory of economic growth". Quarterly Journal of Economics . 70 (1): 65–94. doi:10.2307/1884513. hdl:10338.dmlcz/143862. JSTOR   1884513.
3. Swan, Trevor W. (November 1956). "Economic growth and capital accumulation". Economic Record. 32 (2): 334–361. doi:10.1111/j.1475-4932.1956.tb00434.x.
4. Solow, Robert M. (1957). "Technical change and the aggregate production function". Review of Economics and Statistics . 39 (3): 312–320. doi:10.2307/1926047. JSTOR   1926047.
5. Haines, Joel D.; Sharif, Nawaz M. (2006). "A framework for managing the sophistication of the components of technology for global competition". Competitiveness Review: An International Business Journal. 16 (2): 106–121. doi:10.1108/cr.2006.16.2.106.
6. Blume, Lawrence E.; Sargent, Thomas J. (2015-03-01). "Harrod 1939". The Economic Journal. 125 (583): 350–377. doi:10.1111/ecoj.12224. ISSN   1468-0297.
7. Besomi, Daniele (2001). "Harrod's dynamics and the theory of growth: the story of a mistaken attribution". Cambridge Journal of Economics. 25 (1): 79–96. doi:10.1093/cje/25.1.79. JSTOR   23599721.
8. Harrod, R. F. (1939). "An Essay in Dynamic Theory". The Economic Journal. 49 (193): 14–33. doi:10.2307/2225181. JSTOR   2225181.
9. Halsmayer, Verena; Hoover, Kevin D. (2016-07-03). "Solow's Harrod: Transforming macroeconomic dynamics into a model of long-run growth". The European Journal of the History of Economic Thought. 23 (4): 561–596. doi:10.1080/09672567.2014.1001763. ISSN   0967-2567.
10. Romer, David (2006). Advanced Macroeconomics. McGraw-Hill. pp. 31–35. ISBN   9780072877304.
11. Baumol, William J. (1986). "Productivity Growth, Convergence, and Welfare: What the Long-Run Data Show". The American Economic Review. 76 (5): 1072–1085. JSTOR   1816469.
12. Barelli, Paulo; Pessôa, Samuel de Abreu (2003). "Inada conditions imply that production function must be asymptotically Cobb–Douglas" (PDF). Economics Letters. 81 (3): 361–363. doi:10.1016/S0165-1765(03)00218-0.
13. Litina, Anastasia; Palivos, Theodore (2008). "Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment". Economics Letters . 99 (3): 498–499. doi:10.1016/j.econlet.2007.09.035.
14. Romer, David (2011). "The Solow Growth Model". Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. ISBN   978-0-07-351137-5.
15. Caselli, F.; Feyrer, J. (2007). "The Marginal Product of Capital". The Quarterly Journal of Economics. 122 (2): 535–68. CiteSeerX  . doi:10.1162/qjec.122.2.535.
16. Lucas, Robert (1990). "Why doesn't Capital Flow from Rich to Poor Countries?". American Economic Review . 80 (2): 92–96
17. Mankiw, N. Gregory; Romer, David; Weil, David N. (May 1992). "A Contribution to the Empirics of Economic Growth". The Quarterly Journal of Economics. 107 (2): 407–437. CiteSeerX  . doi:10.2307/2118477. JSTOR   2118477.
18. Klenow, Peter J.; Rodriguez-Clare, Andres (January 1997). "The Neoclassical Revival in Growth Economics: Has It Gone Too Far?". In Bernanke, Ben S.; Rotemberg, Julio (eds.). NBER Macroeconomics Annual 1997, Volume 12. National Bureau of Economic Research. pp. 73–114. ISBN   978-0-262-02435-8.
19. Breton, T. R. (2013). "Were Mankiw, Romer, and Weil Right? A Reconciliation of the Micro and Macro Effects of Schooling on Income" (PDF). Macroeconomic Dynamics. 17 (5): 1023–1054. doi:10.1017/S1365100511000824. hdl:10784/578.
20. Breton, T. R. (2013). "The role of education in economic growth: Theory, history and current returns". Educational Research. 55 (2): 121–138. doi:10.1080/00131881.2013.801241.
21. Barro, Robert J.; Sala-i-Martin, Xavier (2004). "Growth Models with Exogenous Saving Rates". Economic Growth (Second ed.). New York: McGraw-Hill. pp. 37–51. ISBN   978-0-262-02553-9.
22. Barro, Robert J.; Sala-i-Martin, Xavier (2004). "Growth Models with Exogenous Saving Rates". Economic Growth (Second ed.). New York: McGraw-Hill. pp. 461–509. ISBN   978-0-262-02553-9.