# 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?

• I suggest rewriting these expressions in less opaque terms s.t. you can clearly see how the population growth rate enters into each equality. That should obviate the effect of a change in the population growth rate.
– 123
Dec 11 '16 at 17:47
• I think you have something missing in your Keynes-Ramsey equation for population growth Dec 13 '16 at 17:16

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 $\tilde{c}>\hat{c}$.

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.

• Thanks! I intuitively found more reasonable that the economy would transition to the new steady state progressively not instantaneously. However, I cannot get a transition path that converges to the new steady state. If I look at the dynamics I see that a decrease in population growth would at first cause the capital to increase, but that would cause a progresive decrease in consumption that will accelerate the increase in capital, eventually diverging from the new steady state. Thus, the only answer I found that fits with the dynamics is a sudden increase in c. Which step I am missing? Dec 12 '16 at 14:46
• Hmmm -- you might be right. It's been ages since I played around with Ramsey-Cass-Koopmans, so I need to think about this more carefully, but I believe the old steady-state does not lie in the region (relative to the new steady-state) that guarantees convergence (to the SS). Dec 12 '16 at 23:26

You can find easily the effect of a decrease by making a comparative static analysis. First, I just write the steady state value of $k$. (Note that you have made a little mistake in your calculation for $k^{SS}$ and in your Euler equation for technical progress, there is no $\theta$ in front of $g$.)

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

After, I plug into the steady state level of consumption and I find ;

$$\left(\frac{\alpha}{\rho+n+\delta+g}\right)^{\frac{\alpha}{1-\alpha}}-n\left(\frac{\alpha}{\rho+n+\delta+g}\right)^{\frac{1}{1-\alpha}}$$

Then, you can easily find the effect of a decrease in population growth by differentiating $k^{SS}$ abd $c^{SS}$ with respect to $n$ and find the effect.

The effect of an increase in $n$ will surely decrease the steady state level of capital but this effect is ambiguous for consumption at steady state.

• I will check my results but I think that maybe we are using different utility functions. I derive my results from the following utility function: $\int_0^\infty e^{- \rho t} \frac {c(t)^{\theta}}{1-\theta} L(t) dt$, where $c(t)$ is the consumption per worker. I get the $\theta$ before the $g$ when I express this utility function in terms of consumption per effective worker Dec 14 '16 at 14:25
• @Haarlem90 The power for $c(t)$ should be $1-\theta$ and not only $\theta$ if you want to use a CRRA utility function. Dec 14 '16 at 22:06