Solow–Swan model

The Solow–Swan model was an extension to the 1946 Harrod–Domar model that dropped the restrictive assumption that only capital contributes to growth (so long as there is sufficient labor to use all capital). 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.[6] 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.[7]

The Solow model is also one of the most widely used models in economics to explain economic growth.[8] Basically, it asserts that outcomes on the "total factor productivity (TFP) can lead to limitless increases in the standard of living in a country."[8]

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.[15][16]

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

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 ${\displaystyle 0<c<1}$) is consumed, leaving a saved share ${\displaystyle s=1-c}$ for investment. This dynamic is expressed through the following differential equation:

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

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 ${\displaystyle y}$, which is a measure for wealth creation:[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.[17]^: 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 ${\displaystyle (n+g)}$. Likewise, it is possible to calculate the steady-state of created wealth ${\displaystyle y^{*}}$ that corresponds with ${\displaystyle k^{*}}$:

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

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.[17]^: 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,[18] 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.[19]

Because the marginal product of capital ${\displaystyle {\frac {\partial Y}{\partial K}}}$ equals the rate of return ${\displaystyle r}$

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

so that ${\displaystyle \alpha }$ is the fraction of income appropriated by capital. Thus, the Solow–Swan model assumes from the beginning that the labor-capital split of income is constant.

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.[23]

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.[24]

Additional evidence for conditional convergence comes from multivariate, cross-country regressions.[25]

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").[7]

• Economic growth
• Endogenous growth theory

• Agénor, Pierre-Richard (2004). "Growth and Technological Progress: The Solow–Swan Model". The Economics of Adjustment and Growth (Second ed.). Cambridge: Harvard University Press. pp. 439–462. ISBN 978-0-674-01578-4.
• Barro, Robert J.; Sala-i-Martin, Xavier (2004). "Growth Models with Exogenous Saving Rates". Economic Growth (Second ed.). New York: McGraw-Hill. pp. 23–84. ISBN 978-0-262-02553-9.
• Burmeister, Edwin; Dobell, A. Rodney (1970). "One-Sector Growth Models". Mathematical Theories of Economic Growth. New York: Macmillan. pp. 20–64.
• Dornbusch, Rüdiger; Fischer, Stanley; Startz, Richard (2004). "Growth Theory: The Neoclassical Model". Macroeconomics (Ninth ed.). New York: McGraw-Hill Irwin. pp. 61–75. ISBN 978-0-07-282340-0.
• Farmer, Roger E. A. (1999). "Neoclassical Growth Theory". Macroeconomics (Second ed.). Cincinnati: South-Western. pp. 333–355. ISBN 978-0-324-12058-5.
• Ferguson, Brian S.; Lim, G. C. (1998). Introduction to Dynamic Economic Models. Manchester: Manchester University Press. pp. 42–48. ISBN 978-0-7190-4996-5.
• Gandolfo, Giancarlo (1996). "The Neoclassical Growth Model". Economic Dynamics (Third ed.). Berlin: Springer. pp. 175–189. ISBN 978-3-540-60988-9.
• Halsmayer, Verena (2014). "From Exploratory Modeling to Technical Expertise: Solow's Growth Model as a Multipurpose Design". History of Political Economy. 46 (Supplement 1, MIT and the Transformation of American Economics): 229–251. doi:10.1215/00182702-2716181. Retrieved 2017-11-29.
• Intriligator, Michael D. (1971). Mathematical Optimalization and Economic Theory. Englewood Cliffs: Prentice-Hall. pp. 398–416. ISBN 978-0-13-561753-3.
• van Rijckeghem Willy (1963) : The Structure of Some Macro-Economic Growth Models : a Comparison. Weltwirtschaftliches Archiv volume 91 pp. 84–100

• Solow Model Videos - 20+ videos walking through derivation of the Solow Growth Model's Conclusions
• Video explanation by Marginal Revolution University
• Java applet where you can experiment with parameters and learn about Solow model
• Solow Growth Model by Fiona Maclachlan, The Wolfram Demonstrations Project.
• A step-by-step explanation of how to understand the Solow Model
• Professor José-Víctor Ríos-Rull's course at University of Minnesota