The Solow-Swan growth model

An extended application of mathematical economics

Another entry in my continuing series on economics, this time with what passes in mathematics for an example.

After several weeks of notation and theory, I thought it would be time to go through a worked example. The best application I’ve encountered involves the theory of exogenous growth (Wiki) by Robert Solow and Trevor Swan. This model predates the variational Ramsey-Cass-Koopmans model (Wiki) which makes basically the same predictions but whose “microfounded” formulation has become the cornerstone of modern macro-economic models. More on that one another time.

The purpose of this example is not to learn anything new about the Solow-Swan model, but to illustrate how the geometric framework gives us some new levers—new variables, constrained in a very specific way—to expand and extend the model. Find the previous post in the series below:

Deutero-canonical coordinates

Exogenous growth

The Solow-Swan model (Wiki) is a model of how capital formation occurs in an economy. I’ll give a simiplified presentation, focused on the math. Of course.

The economy has two factors of production, labour L and capital K, and a production function of Cobb-Douglas form F(K,L)=K^α L^(1-α). Assuming a fraction σ of income is saved/invested to form new capital¹, capital depreciates at rate δ, and writing k=K/L for capital intensity, suggests its rate of change satisfies the differential equation²:

\(\dot{k} = \sigma k^\alpha - \delta k\)

The argument goes that equilibrium is reached when this change vanishes, so

\(k^* =\left( \frac{\sigma}{\delta}\right)^{\left( \frac{1}{1-\alpha} \right)}\)

This determines a “balanced growth equation” or “golden rule savings rate (Wiki)”. And the differential equation is solvable, so we can in principle determine how the economy approaches that equilibrium whatever its initial state.

Deutero-canonical version

But with our we can do more! We can back out the generating function G satisfying³:

\(\dot{q} = \frac{\partial G}{\partial p}, \quad \dot{p} = -\frac{\partial G}{\partial q} -p \frac{\partial G}{\partial s}, \quad \dot{s} = p\frac{\partial G}{\partial p}-G \qquad (*)\)

Where now k plays the role of the state variable q; and its price (or co-state variable) p and “value” s are to be determined. What is s you ask? Why, Solow Money of course!

I couldn’t resist, I hope he would approve.

More seriously, it’s not necessarily money but it has the same units utility would have. Then from \dot{k}=∂G/∂p, we can guess the general form for G(s,k,p) to find

\(G(s,k,p) = p(\sigma k^\alpha - \delta k) + g(s,k)\)

Where g(s,k) is some “gauge” function independent of p. The other two flow relations then give us:

\(\dot{p} = -p(\alpha \sigma k^{\alpha-1} - \delta + g_s) -g_k ; \quad \dot{s} = -g(s,k) ; \quad \text{and remember } \; \dot{G} = -g_s G\)

where we have written subscripts for partial derivatives of g. We see G decays to 0 exponentially at rate gₛ(s,k), and p also decays exponentially though not to 0. We can determine its limit from \dot{p}=0, and then insert k* we found before, to see that

\(p^* = \frac{-g_k}{\alpha \sigma k^{\alpha-1} -\delta +g_s} \to \frac{g_k^*}{(1-\alpha)\delta - g_s^*} \quad \text{since } (k^*)^{(\alpha-1)}=\delta / \sigma\)

and G(s,k*,p) = g(s,k*)=0⁴. We see imposing dynamics on k results in constraining the dynamics of its associated price and system value. Whatever we choose for g(s,k) will not affect the conclusions of Solow-Swan theory, but it can “gauge” the behavior of the equilibrium price of capital and the rate at which the system approaches its equilibrium. Economic considerations might be used to determine gₛ and the limit p* which would allow understanding the model in far-from-equilibrium economies.

1. Wikipedia calls this s but I’m using that name for another variable, so go Greek.

2. We ignore population growth n and technology improvement g for the moment. These are economically important but exogenous (hence the name)—they are fixed parameters that don’t affect the model. They can be considered as additions to depreciation δ.

3. Note that this model doesn’t explicitly state my postulates, but I work on the assumption that they are implied. Making assumptions makes you an economist, right?

4. That means there is a limit s* as well.