The Solow Model

The Solow-Swan model of economic growth is the basic framework used by economists to model economic growth.  Here are some notes about it.

The Basic Ideas

Flows, not stocks: the Solow model is a very simple flow model, a model of the rate of production of something, specified as the quantity produced in one time period. Stocks enter the model only indirectly and implicitly. The model uses two equations, one for production, and one describing how one of the two free variables of the production function, "capital", changes. It also uses several rather ad hoc and unrealistic auxiliary assumptions. (But the point of a model is to be simple, and just realistic enough to be useful, so these aren't necessarily problematic.)

(To jump ahead a bit, although the Solow model appears to talk about stocks of capital and workers, what it actually talks about or assumes is the flow of services that these stocks provide when used--generally called "capital" and "labour". It is assumed that both stocks are fully used at all times, so the flow of services can be measured by measuring the stocks, but sometimes it pays to be clear about the difference between capital-the-stock and capital-the-flow.)

The rate of production, usually given as $Y$, is often referred as "the quantity produced", with "...per unit time" usually omitted.

$Y$ depends on $K$ and $L$, the rate of flow of services from the stock of capital, and the rate of flow of labour from the stock of workers.

In the basic model, only the magnitudes of the available (services of) capital ($K$) and of labour ($L$)  constrain $Y$. Nothing else does. In particular, the basic model ignores the role of mechanical and chemical work done by energy sources, and scarcity of raw materials. Various extensions attempt to incorporate thermodynamic work and resource scarcity.  For now, we'll stick to the basics.

The Production Function, part I

 $Y$ is a function of $K$ and $L$, which are themselves considered as varying over time:  Capital is a function $K(t)$, and Labour $L(t)$. So we have $$Y(t) = f(K(t), L(t))$$ for some function $f$. ($Y$ is probably from yield, once a synonym of harvest.) $Y$ is thus a function of two functions of time: the factors of production, capital and labour.

 Correction Factor $B$

To allow for the fact that in olden times, a given quantity of both capital and workers produced less than the same quantity of both produces now, a third function is introduced, normally written $B(t)$ or $A(t)$, and called "the level of technology at time $t$", although what it actually is, is problematic. In the equation, $B$ is dimensionless, just a multiplier.

So  we have $$Y(t) = f(K(t), L(t), B(t)).  For no very discernible reason (TODO: consult the early literature), economists like to bind the multiplier to the $L$ term, asserting that increasing "the level of technology" increases the effective rate of labour supplied by a given stock of workers. In this form the $B$ is changed to an $A$. (Actually, this assumption makes the maths easier, which is much less of a concern in these days of Maxima, R and Octave than it was in 1957, and it's not a great reason anyway.)

Summary so far

• Production equation $Y(t) = f(K(t), A(t)L(t))$

• $Y(t)$:  The rate of flow of services produced at time $t$.  It depends only on $K$ and $L$.
• everything else is assumed not to be scarce -- i.e., to be available in whatever quantity is desired.
• $K(t)$: The flow  of services from capital goods at time $t$.
• $L(t)$: The flow of services from the stock of workers at time $t$.
• $B(t)$ or $A(t)$: a correction function to make things come out right.
• Assumption that "the level of technology" increases the flow of labour services.

From now on we'll drop the $(t)$ notation where it can be inferred.

Change in Capital

In the Solow model capital is increased or replaced by saving a proportion of Y, the flow of services produced, and using that to increase the stock of capital and therefore the flow of capital services used. Meanwhile, the flow of capital services is reduced by depreciation of the stock of capital.  The fraction of output saved, $s$, and the rate at which capital depreciates, $\delta$, are both assumed to be fixed constants, mainly for mathematical convenience.  The stock of capital, and hence the flow of services from it, is increased by saving, and decreased by depreciation. In one time period the change in capital $\Delta K$ is thus  $$\Delta K = sY - \delta K .$$ Two further points here. $K$ is assumed to change continuously so in the limit we have $$dK/dt = sY - \delta K ,$$ and $Y$ depends on $K$:$$dK/dt = sf(K, AL) - \delta K .$$

To be continued

OK.  Later posts will have some discussion about $f$; the assumptions, the economic model behind the equations; and exposition, discussion of the key result of the model; and discussion of extensions.