I am trying to understand the way Smith arrived from (1) and (2) to (6) and (8).
Could anyone give me a hint? Thank you!
It is well known that the dynamics per capita consumption $c$ and capital $k$ are determined by the following pair of differential equations: \begin{eqnarray} \dot{k} &=& k^\alpha - \delta k - c \tag{1} \\ \frac{{\dot c}}{c} &=& \sigma(\alpha k^{\alpha - 1} - \delta - \rho) \tag{2} \end{eqnarray} A Solution
The system of equations (1)-(2) is dauntingly nonlinear. To approach a solution, define the capital-output ratio (a Bernoulli transformation) by $z = k^{1-\alpha}$ and the consumption-capital ratio by $x = c/k$. Using these transformations the system can be rewritten as \begin{eqnarray} \dot{z} &=& (1-\alpha)[1 - (\delta + x)z] \tag{6} \\ \frac{{\dot x}}{x} &=& \frac{\sigma \alpha - 1}{z} + \delta (1 - \sigma) - \rho\sigma + x \tag{7} \end{eqnarray} The Bernoulli transformation converts the capital accumulation equation [Equation (1)] into the linear, albeit non-autonomous, differential equation in Equation (6). Despite this simplification, the system still does not allow an analytical solution. Equation (7) is still non-linear.
Suppose, however, that $\sigma = 1/\alpha$. In that case $z$ disappears from Equation (7), and the system becomes recursive: Equation (7) reduces to $$ \frac{\dot{x}}{x} = - \frac{\delta(1- \alpha) \rho}{\alpha} + x \tag{8} $$ while $z$ evolves according to Equation (6), given the forcing process $x$ determined in Equation (8). This is a simple, autonomous logistic equation.
