Outsourcing global stability in the Solow growth model

Aug. 17, 2014

In the Solow growth model in economics, the dynamics of the capital-labor ratio (\(k\)) is described by the differential equation \[\frac{dk}{dt} = sf(k) - nk\] where \(s\) is a constant between \(0\) and \(1\), \(n\) is a positive constant and \(f\) is a twice-differentiable function with the following properties

  1. \(f(0)=0\)
  2. \(f'(k)>0\)
  3. \(f''(k)<0\)
  4. \(\lim_{k \to 0} f'(k) = \infty\)
  5. \(\lim_{k \to \infty}f'(k) = 0\)

These assumptions guarantee the existence of an equilibrium \(k^*\) for the differential equation. Moreover \(\dot k>0\) for \(k<k^*\) and \(\dot k<0\) for \(k>k^*\).

Every year, being in a hurry to finish the Solow model and move on to more interesting things, I give my students the following story: “since \(k\) is increasing to the left of \(k^*\) and decreasing to the right of it, it must be the case that regardless of your starting point the trajectory of \(k\) will converge to \(k^*\) as time goes to infinity”. That is, \(k^*\) is a globally asymptotically stable equilibrium.

But then my mathematical conscience makes me point out that while this argument is morally right, it is mathematically incomplete. Say we start off to the left of \(k^*\). Then we know from the sign of \(\dot k\) that \(k\) must increase as time passes. But what stops it from increasing ever more slowly as time passes and ending up converging to some point strictly to the left of \(k^*\)? We need to make use of the properties of \(f\) and the solution trajectory itself to argue that this cannot happen.

Having said this, I now have to worry that some smart student will actually ask me to write out a full proof. The standard growth theory texts I have at hand don’t prove this, but here is a one-line proof if you are willing to outsource the difficulty to someone else:

Use \(V(k)=(k-k^*)^2\) as your Lyapunov function.

For the details that we thus outsource, see for example Theorem 2.3 of Perko’s Differential Equations and Dynamical Systems, 3rd ed. or the theorem in Section 9.2 of Hirsch, Smale and Devaney’s Differential Equations, Dynamical Systems & an Introduction to Chaos, 2nd ed.