Closed-form solution of the CARA Ramsey Abel equation

Derive a closed-form solution for the second-kind Abel ordinary differential equation governing the optimal saving function in the neoclassical growth model with constant absolute risk aversion, Cobb–Douglas production, and a generic discount rate satisfying \(\rho>n\), beyond the special case \(\rho=n\).

Background

For the neoclassical growth model with CARA utility and Cobb–Douglas production, the optimal saving function satisfies the first-order equation $s'(k)s(k)=(r(k)-n)s(k)-(r(k)-\rho)/\gamma$, together with the terminal condition that saving converges to zero at the steady state. The paper identifies this equation as an Abel ordinary differential equation of the second kind.

The paper obtains an explicit Lambert $W$-function solution only in the auxiliary case $\rho=n$, then uses smooth dependence on $\rho$ to show that this solution approximates the true policy functions as $\rho\to n^+$. A closed-form solution for the generic economically relevant case remains unresolved because the specific Abel equation does not have the structure required by the known solvable subclasses.