<p>I suppose you are speaking of the standard Solow Growth model.</p>
<p>Yes, in this case, the economic system will come back to the original steady state equilibrium. This is a consequence of the <strong>stability</strong> of the steady state equilibria in the standard Solow model, that is, if the system  is not on its steady state equilibrium, there are forces that lead it to equilibrium.</p>
<p>It can be shown graphically that a <strong>temporary</strong> increase of saving rate, with the saving rate first increasing and then falling back to the original level, causes the system <strong>to go back to the original steady state</strong>.</p>
<p>The fundamental equation in the standard Solow model (in continuous time) is:</p>
<p><span class="math-container">$$\dot k= sf(k) - (n+d)k,$$</span></p>
<p>where <span class="math-container">$k$</span> is capital/labour ratio, <span class="math-container">$f(k)$</span> is the intensive production function, <span class="math-container">$n$</span> the rate of growth of population, <span class="math-container">$d$</span> depreciation, <span class="math-container">$\dot k$</span> the derivative of <span class="math-container">$k$</span> with respect to time.</p>
<p>When <span class="math-container">$\dot k=0$</span> the system is on its steady state path, where <span class="math-container">$k$</span> is constant and it can be shown that all per capita quantities are constant.</p>
<p>This dynamics can be represented graphically as usual:</p>
<p><a href="https://i.sstatic.net/RvoDN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/RvoDN.png" alt="enter image description here" /></a></p>
<p>Suppose the system begins at the steady state <span class="math-container">$E$</span>, with a steady state of capital per worker of <span class="math-container">$k^*$</span>. If <span class="math-container">$s$</span> increases to the new level <span class="math-container">$s'$</span>, the <span class="math-container">$sf(k)$</span> curve shifts upward (the red curve), and we have the new steady state <span class="math-container">$E'$</span>. The dynamics of <span class="math-container">$k$</span>  is now  represented by the red  arrows, pointing to the new steady state level of <span class="math-container">$k$</span>, that is <span class="math-container">$k^{**}.$</span>  During the process of adjustment there is a transitional dynamics of <span class="math-container">$k$</span>, that is <span class="math-container">$k$</span> increases, but  this dynamics will stop when <span class="math-container">$k$</span> has reached its new steady, long run, constant value <span class="math-container">$k^{**}$</span>.</p>
<p>But the saving rate, at a subsequent time, falls back to the previous value. The <span class="math-container">$s'f(k)$</span> curve shifts downward and becomes again the blue curve.  So, the opposite  situation occurs: now the steady state is again <span class="math-container">$E$</span>, and the dynamics of <span class="math-container">$k$</span> is described by the blue arrows: the blue arrows point to <span class="math-container">$k^*$</span>, there is a transitional dynamics toward it, which stops when the system has again reached the old steady state <span class="math-container">$E$</span>.</p>
<p>The system will therefore come back and remain in the old steady state equilibrium.<span class="math-container">$^1$</span></p>
<hr />
<p><span class="math-container">$^1$</span> <sup> For a similar, more detailed, exposition, but with a different graph, see Acemoglu, <em>Modern Economic Growth</em>, chap. 2.</sup></p>

CC BY-SA 4.0
2022-11-25T20:45:13.553Z
2022-11-27T13:46:28.227Z
2022-11-27T13:46:28.227Z
BakerStreet
<p>If there is a increase in saving rate and fall back to original level afterwards, what is the effect on steady state in the short run and long run?
I am confused about this is because I thought increase in saving rate would cause a increase in gross investment and lead to a new steady state, but the saving rate falling back to original level and I am not sure whether will return to the original or not.</p>

<macroeconomics><economic-growth><saving><steady-state>
The effect of saving rate on steady state
sharonzj f
# The effect of saving rate on steady state

(Asked by: username_0 on 2022-11-25)

If there is a increase in saving rate and fall back to original level afterwards, what is the effect on steady state in the short run and long run?
I am confused about this is because I thought increase in saving rate would cause a increase in gross investment and lead to a new steady state, but the saving rate falling back to original level and I am not sure whether will return to the original or not.

**Tags:** macroeconomics, economic-growth, saving, steady-state

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## 1 answers

### Answer by username_1 on 2022-11-25. Score: 3

I suppose you are speaking of the standard Solow Growth model.

Yes, in this case, the economic system will come back to the original steady state equilibrium. This is a consequence of the **stability** of the steady state equilibria in the standard Solow model, that is, if the system is not on its steady state equilibrium, there are forces that lead it to equilibrium.

It can be shown graphically that a **temporary** increase of saving rate, with the saving rate first increasing and then falling back to the original level, causes the system **to go back to the original steady state**.

The fundamental equation in the standard Solow model (in continuous time) is:

$$\dot k= sf(k) - (n+d)k,$$

where $k$ is capital/labour ratio, $f(k)$ is the intensive production function, $n$ the rate of growth of population, $d$ depreciation, $\dot k$ the derivative of $k$ with respect to time.

When $\dot k=0$ the system is on its steady state path, where $k$ is constant and it can be shown that all per capita quantities are constant.

This dynamics can be represented graphically as usual:

[![enter image description here](https://i.sstatic.net/RvoDN.png)](https://i.sstatic.net/RvoDN.png)

Suppose the system begins at the steady state $E$, with a steady state of capital per worker of $k^\*$. If $s$ increases to the new level $s'$, the $sf(k)$ curve shifts upward (the red curve), and we have the new steady state $E'$. The dynamics of $k$ is now represented by the red arrows, pointing to the new steady state level of $k$, that is $k^{\*\*}.$ During the process of adjustment there is a transitional dynamics of $k$, that is $k$ increases, but this dynamics will stop when $k$ has reached its new steady, long run, constant value $k^{\*\*}$.

But the saving rate, at a subsequent time, falls back to the previous value. The $s'f(k)$ curve shifts downward and becomes again the blue curve. So, the opposite situation occurs: now the steady state is again $E$, and the dynamics of $k$ is described by the blue arrows: the blue arrows point to $k^\*$, there is a transitional dynamics toward it, which stops when the system has again reached the old steady state $E$.

The system will therefore come back and remain in the old steady state equilibrium.$^1$

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$^1$  For a similar, more detailed, exposition, but with a different graph, see Acemoglu, *Modern Economic Growth*, chap. 2.


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