The Solow growth model
A short introduction and graphical simulation
Huub Meijers, March 2012 (this page is based on demonstration materials provided by Richard Foltyn on the Wolfram website)
This short note provides two versions of the Solow growth model, often referred to as the Solow  Swan model.The first model is without technological change and wihtout growth of the population. After that, a second version introduces both exogenous technological change and allows for growth of the population.
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:
(1)
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:
(2)
and since investments (I) equals savings (S) the amount of investments is also a constant fraction (s) of output :
(3)
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:
(4)
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. If the economy is below the steady state savings will exceed depreciation and thus the output per worker will increase. If the economy is above its steady state the amount of savings is below to amount of depreciation such that the capital per worker will decrease. This implies that the steady state in the Solow model is a stable equilibrium.
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 goldenrule 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 goldenrule capital stock.
The interactive graph is created by using Mathematica from Wolfram using a CDF technology
(Computable Document Format). Though the graph itself is fairly small, the CDF player is rather large and Wolfram
asks users to provide some details (e.g. email address). However, you need the player to be able to see and play with the interactive graphs.
Model with technological change and with growth of the population
The model is basically the same as the previous except that the technology parameter A is now growing at a rate g and the population (labour force) is allowed to grow at a rate n.
References
R.M.Solow, 1956, "Contribution to the Theory of Economic Growth," The Quarterly Journal of Economics, 70, pp.65–94. T.W.Swan, 1956, "Economic Growth and Capital Accumulation," Economic Record, 32.
