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The Solow growth model A short introduction and graphical simulation
Huub Meijers, March 2012
Basic model without technological change The Solow model is based on three simple equations: the production function linking capital and labour as inputs to output, the savings function linking output to savings and a dynamic capital accumulation function linking the existing (current) capital stock, depreciation and investment to an updated capital stock in next period. The production function used here is a linear homogeneous Cobb - Douglas production function and reads as:
where Y is output (value added), L is labour input, K is capital input and A is a technology parameter. This function can be rewritten in terms of output per worker (so dividing both sides by L) as:
and since investments (I) equals savings (S) the amount of investments is also a constant fraction (s) of output :
Finally we have to update the capital stock where it is assumed that a fraction (δ) of the capital stock depreciates because of wear and tear such that the capital stock at time t + 1 equals the capital stock at time t minus δ times the capital stock at time t plus the amount of investment at time t:
The Solow growth model can now be presented graphically by three lines: the output per worker, the investments per worker and the depreciation per worker. If the investments per worker is below the depreciation per worker the amount of capital per worker will decrease over time. On the other hand if the investments per worker is above the depreciation per worker the amount of capital per worker will increase over time. If both are equal the capital per worker wil remain constant and this situation is called the steady state. So for a given savings rate the economy will tend to the steady state. If the savings rate increases, the investments per worker increase but the amount of consumption will drecrease. Increased investment will however lead to more output and thus to more consumption in the future. But if the savings rate is increased even further and reaches one, the amount of consumption per worker will decrease towards zero since all output is saved. In this situation the amount of capital per worker will be excessive. This implies that at very low values of the savings rate the consumption will be very low because of the lack of capital per worker whereas at very high levels of the savings rate the amount of consumption per worker will be low because of an excissive amount of capital per worker. The optimal savings rate will be somewhere in between and this level is called the golden-rule level of capital. In the diagram below a baseline solution of the model is provided as well as the state that is adjusted by moving the sliders. k* stands for the steady state capital stock per worker, y* and c* are the steady state output per worker and consumption per worker, respectively. kg denotes the golden-rule capital stock. The interactive graph is created by using Mathematica from Wolfram using a CDF technology (Computable Document Format). The initial CDF player could be used as plug-in within all popular browsers. However, since browser support less and less plug-ins (mainly through the development of HTML5) this is not possible any more. Instead, you can download the model without and with technological change also download the stand alone CDF player to run the examples on a PC.
The model without technological change
R.M.Solow, 1956, "Contribution to the Theory of Economic Growth," The Quarterly Journal of Economics, 70, pp.65–94. |
