Effect in the Ramsey model of a decrease in population growth In the basic Ramsey model with technological growth, assuming that the economy is in a steady state, how would a sudden decrease in the population growth rate $n$ impact the steady states value of consumption and capital?

I obtain the following dynamics for the model:

$\dot{k} = f(k) - c - (\delta + n + g)k$

$\frac{\dot{c}}{c} = \frac{1}{\theta}[f'(k) - \delta - \rho -\theta g] $

where $\delta$ is the depreciation rate, $g$ the rate of technical progress, $\theta$ the risk aversion factor, $\rho$ the discout factor, $k$ the capital per effective worker, $f(k)$ the production per effective worker, $c$ the consumption per effective worker and $ f(k) = k^\alpha $

The steady state values, ($\dot{k}=0, \dot{c}=0$):

$\hat{c} = \hat{k}^\alpha - (\delta + n + g)\hat{k}$

$\hat{k} = \left( \frac{\delta + \rho + \theta g}{\alpha} \right)^{\frac{1}{1-\alpha}}$

I believe that there would be a sudden increase in the steady state consumption that would persist in the long run and no change in the capital. Is it right?
Let $\left(\hat{c},\hat{k}\right)$ and $\left(\tilde{c},\tilde{k}\right)$ be the old and new steady-state levels of $c$ and $k$ respectively.

You make the correct observation that $\tilde{k}=\hat{k}$ and $\tilde{c}&gt;\hat{c}$.

However, starting from the steady-state $\left(\hat{c},\hat{k}\right)$, and experiencing an unexpected negative shock to the population growth rate $n$, it is easy to see from the equations determining equilibrium dynamics that the economy does not instantaneously transition to the new steady-state $\left(\tilde{c},\tilde{k}\right)$.

It would be a good exercise for you to work out what the qualitative features of the transition path from $\left(\hat{c},\hat{k}\right)$ to $\left(\tilde{c},\tilde{k}\right)$ look like after the population growth shock.