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The model was developed independently by Robert Solow and Trevor Swan in 1956,^{[2][3][note 1]} and superseded the Keynesian Harrod–Domar model.

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.

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]}

**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]}

**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]}

- 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).

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.

- Given a fixed stock of labor, the impact on output of the last unit of capital accumulated will always be less than the one before.
- Assuming for simplicity no technological progress or labor force growth, diminishing returns implies that at some point the amount of new capital produced is only just enough to make up for the amount of existing capital lost due to depreciation.
^{[1]}At this point, because of the assumptions of no technological progress or labor force growth, we can see the economy ceases to grow. - Assuming non-zero rates of labor growth complicate matters somewhat, but the basic logic still applies
^{[2]}– in the short-run, the rate of growth slows as diminishing returns take effect and the economy converges to a constant “steady-state” rate of growth (that is,*no*economic growth per-capita). - Including non-zero technological progress is very similar to the assumption of non-zero workforce growth, in terms of “effective labor”: a new steady state is reached with constant output per
*worker-hour required for a unit of output*. However, in this case, per-capita output grows at the rate of technological progress in the “steady-state”^{[3]}(that is, the rate of productivity growth).

**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.

References

**^***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.*- ^ Jump up to:
^{a}^{b}^{c}*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. - ^ Jump up to:
^{a}^{b}*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. **^***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.- ^ Jump up to:
^{a}^{b}*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. **^***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.- ^ Jump up to:
^{a}^{b}*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. **^***Harrod, R. F. (1939). “An Essay in Dynamic Theory”. The Economic Journal.***49**(193): 14–33. doi:10.2307/2225181. JSTOR 2225181.**^***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.**^***Romer, David (2006). Advanced Macroeconomics. McGraw-Hill. pp. 31–35. ISBN 9780072877304.***^***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.**^***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.**^***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.- ^ Jump up to:
^{a}^{b}*Romer, David (2011). “The Solow Growth Model”. Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. ISBN 978-0-07-351137-5.* **^***Caselli, F.; Feyrer, J. (2007). “The Marginal Product of Capital”. The Quarterly Journal of Economics.***122**(2): 535–68. CiteSeerX 10.1.1.706.3505. doi:10.1162/qjec.122.2.535.**^***Lucas, Robert (1990). “Why doesn’t Capital Flow from Rich to Poor Countries?”. American Economic Review.***80**(2): 92–96**^***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 10.1.1.335.6159. doi:10.2307/2118477. JSTOR 2118477.**^***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.*- ^ Jump up to:
^{a}^{b}*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. **^***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.**^***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.***^***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.*