Details

Formally, the main aspects of the Solow–Swan model can be derived from the differential equation that describes capital accumulation: . Capital (per effective labor) accumulates if the proportion of output saved (and thus invested) is higher than the break-even investment, defined as the amount of capital needed to cover depreciation , population growth , and technological progress . In the steady state, capital per effective labor is constant by definition and hence . The standard Cobb–Douglas production function is used in this Demonstration, so its intensive form is defined as , where is the elasticity of capital.

It is evident from the model dynamics that only a change in the rate of technological progress has any influence on the long-run growth rate of per-capita output and consumption. For any other parameter change, the growth rate eventually converges back to the original value of .

References:

[1] R. J. Barro and X. Sala-i-Martin, "Growth Models with Exogenous Saving Rates (The Solow–Swan Model)," Economic Growth, 2nd ed., Cambridge, MA: MIT University Press, 2004 pp. 14-58.

[2] D. Romer, "The Solow Growth Model," Advanced Macroeconomics, 3rd ed., New York: McGraw–Hill, 2006.

[3] R. M. Solow, "Contribution to the Theory of Economic Growth," The Quarterly Journal of Economics, 70(1), 1956 pp. 65–94.

[4] T. W. Swan, "Economic Growth and Capital Accumulation," Economic Record, 32, 1956.