Effect of an increase/decrease of ρ or θ on consumption per effective labour and on capital per effective labour? And on their steady-state values?

In a Ramsey-Cass-Koopmans framework, having defined ρ as the rate of time preference, I was wondering what is the effect of a variation in the two parameters on consumption per effective labour and on capital per effective labour and on their steady state values.

Assuming that $$f$$ is the Cobb-Douglas production function in per capita terms, then is the steady-state equilibrium consumption $$c^*$$ and capital $$k^*$$ uniquely determined by $$f'(k^*) = \delta + \rho \quad \text{and} \quad c^* = f (k^*) - (n + \delta) k^*$$ with depreciation rate $$\delta$$, time preference $$\rho$$, and growth rate $$n$$.
Since $$f$$ is assumed to be Cobb-Douglas it holds $$f' > 0$$ and $$f'' < 0$$. Hence is $$f$$ strictly monotone increasing and $$f'$$ strictly monotone decreasing. Thus, for $$\tilde \rho$$ with $$\tilde \rho < \rho$$, we have $$f'(k^*) = \delta + \rho \quad \text{and} \quad f'(\tilde k^*) = \delta + \tilde \rho,$$ and since $$f'$$ is strictly monotone decreasing we have that $$\tilde k^* > k^*$$ holds. By usind that $$f$$ is strictly monotone increasing we obtain by $$c^* = f (k^*) - (n + \delta) k^* \quad \text{and} \quad \tilde c^* = f (k^*) - (n + \delta) k^*$$ that $$\tilde c^* > c^*$$ holds, too.
• @tryingtogetsmth I accidentally described $f'$ as monotone increasing at the very end which as been fixed. Clearly $f'$ is strictly decreasing, and, thus, the inverse is strictly decreasing, too, i.e., for $\tilde \rho < \rho$ we have that $\tilde k^* > k^*$. You have to see $k^*$ as function in dependency of $\rho$, i.e., $k(\rho) = f'^{-1}(\delta + \rho)$ Jul 1, 2022 at 9:22