# examples/SolowSwan

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Simple economic growth model

$$\begin{array}{rcl} Y & = & K^{ \alpha } \cdot A \cdot L^{ \left(1 - \alpha\right) } \\ L & = & L_{ 0 } \cdot e^{ n \cdot t } \\ A & = & A_{ 0 } \cdot e^{ g \cdot t } \\ \frac{ d K }{ dt } & = & \left(\left(1 - c\right) \cdot Y - \delta \cdot K\right) \\ \end{array}$$
Variable Type Description Value (default) Datatype
$$Y$$ free total production  Real
$$K$$ free capital factor  Real
$$L$$ free labor factor  Real
$$A$$ free labor-augmentation factor  Real
$$L_{ 0 }$$ parameter initial labor $$1$$ Real
$$A_{ 0 }$$ parameter initial labor augmentation $$1$$ Real
$$\alpha$$ parameter elasticity of output with respect to capital (0 to 1)  Real
$$n$$ parameter labor (population) growth rate  Real
$$g$$ parameter augmentation growth rate  Real
$$\delta$$ parameter rate of capital deprecation  Real
$$c$$ parameter consumption vs. investment fraction (0 to 1)  Real

The Solow–Swan model is an exogenous growth model, 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 it is a neoclassical aggregate production function, usually of a Cobb–Douglas type, which enables the model "to make contact with microeconomics". The model was developed independently by Robert Solow and Trevor Swan in 1956, and superseded the post-Keynesian Harrod–Domar model. 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 savings rate—see the Ramsey–Cass–Koopmans model.

## 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.

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:

• Education policy
• Institutional arrangements
• Free markets internally, and trade policy with other countries.

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

If productivity growth were associated only with high technology then the introduction of information technology should have led to a noticeable productivity acceleration over the past twenty years; but it has not: see: Solow computer paradox. Instead world productivity appears to have increased relatively steadily since the 19th century.

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").

## References

The page body text came from Wikipedia.

## Modelica Code

model SolowSwan
"Simple economic growth model"
Real Y "total production";
Real K "capital factor";
Real L "labor factor";
Real A "labor-augmentation factor";
parameter Real L_0=1  "initial labor";
parameter Real A_0=1  "initial labor augmentation";
parameter Real alpha "elasticity of output with respect to capital (0 to 1)";
parameter Real n "labor (population) growth rate";
parameter Real g "augmentation growth rate";
parameter Real delta "rate of capital deprecation";
parameter Real c "consumption vs. investment fraction (0 to 1)";
equation
Y = (((K ^ alpha) * (A * L)) ^ (1 - alpha));
L = ((L_0 * e) ^ (n * t));
A = ((A_0 * e) ^ (g * t));
der(K) = (((1 - c) * Y) - (delta * K));
end SolowSwan;


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