Growth and long-run stability (Q1096071)

The authors consider a scalar equation \(y'=f(y)\) where f is \(C^ 1\) and \(f>0\). Let \(\phi\) (t) be a fixed solution and let \(\psi (t)=\phi (t+h)\) be an arbitrary solution. The stability of f is called strong absolute if \(\lim_{t\to \infty}| \psi (t)-\phi (t)| =0\). Seven other kinds of stability were introduced. Conditions of f leading to each kind of stability (or instability) were obtained. Next, eight types of speed of growth of \(\phi\) were defined. For example, \(\phi\) is said to have unbounded growth if \(\phi '>0\) and \(\lim_{t\to \infty}\phi (t)=\infty\). The two concepts are roughly related by: the more rapid the growth, the less is the degree of stability. (See precise statement in the paper.) These results are then applied to the multiplier-accelerator growth model of Harrod and the neoclassical growth model of Solow in Economics.