<p>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.</p>

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

<p>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 $\left(\tilde{c},\tilde{k}\right)$.</p>

<p>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.</p>

CC BY-SA 3.0
2016-12-12T14:46:59.383Z
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?
Haarlem90
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2016-12-12T23:26:30.450Z
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).
Theoretical Economist
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2016-12-12T00:06:54.337Z
2016-12-12T02:51:44.850Z
2016-12-12T02:51:44.850Z
Theoretical Economist
<p>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 $g$.)</p>

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

<p>After, I plug into the steady state level of consumption and I find ;</p>

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

<p>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. </p>

<p>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. </p>

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2016-12-14T14:25:30.837Z
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
Haarlem90
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2016-12-14T22:06:00.280Z
@Haarlem90 The power for $c(t)$ should be $1-\theta$ and not only $\theta$ if you want to use a CRRA utility function.
optimal control
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2016-12-14T12:02:07.737Z
2016-12-14T12:50:12.617Z
2016-12-14T12:50:12.617Z
optimal control
<p>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?</p>

<p>I obtain the following dynamics for the model:</p>

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

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

<p>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 $</p>

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

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

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

<p>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?</p>

CC BY-SA 3.0
2016-12-11T17:47:39.777Z
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
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2016-12-13T17:16:44.367Z
I think you have something missing in your Keynes-Ramsey equation for population growth
optimal control
<macroeconomics>
Effect in the Ramsey model of a decrease in population growth
Haarlem90
# Effect in the Ramsey model of a decrease in population growth

(Asked by: username_0 on 2016-12-11)

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?

**Tags:** macroeconomics

> **Comments:**
> **username_1**: 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.  
> **username_2**: I think you have something missing in your Keynes-Ramsey equation for population growth  

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## 2 answers

### Answer by username_3 on 2016-12-12. Score: 3

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.

> **Comments:**
> **username_0**: 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?  
> **username_3**: 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).  

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### Answer by username_2 on 2016-12-14. Score: 2

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.

> **Comments:**
> **username_0**: 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  
> **username_2**: @username_0 The power for $c(t)$ should be $1-\theta$ and not only $\theta$ if you want to use a CRRA utility function.  

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