<p>The story in the other answer is not fundamentally wrong but incomplete and bit inaccurate.</p>
<p>Saving does actually affect capital stock through investment in Solow model (assuming based on the Solow tag that is the model you want intuition for), but there is more complexity to it.</p>
<p><strong>What is saving</strong></p>
<p>Investment is actually equal not just to private saving but both private and public saving since saving is actually defined as (following Blanchard et al Macroeconomics: a European Perspective pp 51):</p>
<p><span class="math-container">$$S \equiv Y - T-C$$</span></p>
<p>Combining this definition with output identity for closed economy <span class="math-container">$Y\equiv C+I+G$</span> gives us:</p>
<p><span class="math-container">$$S = I +T -G$$</span></p>
<p>However, it is often convenient in Solow model to just ignore government by assuming <span class="math-container">$G=T$</span>, in which case <span class="math-container">$S+I$</span>.</p>
<p><strong>Why does saving rate increase capital stock:</strong></p>
<p>Next the way how savings rate affects capital accumulation is depicted in the picture I took from Romer Advanced Macroeconomics 4th ed pp 19.</p>
<p><a href="https://i.sstatic.net/GDhM8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GDhM8.png" alt="enter image description here" /></a></p>
<p>As you can see the intuition really works this way. Here <span class="math-container">$s$</span> is the savings rate and <span class="math-container">$f(k)=Y$</span>. Higher <span class="math-container">$s$</span> means that savings increases, for example if <span class="math-container">$Y=100$</span> with <span class="math-container">$s=0.5$</span>, <span class="math-container">$S=50$</span> and with <span class="math-container">$s=0.7$</span>, <span class="math-container">$S=70$</span>. Then if we assume away government indeed <span class="math-container">$S=I$</span> so <span class="math-container">$I=70$</span>.</p>
<p>This is why the evolution of capital accumulation (per unit of effective worker) is given by:</p>
<p><span class="math-container">$$\dot{k}(t) = sf (k(t)) − (n + g +\delta)k(t)$$</span></p>
<p>where actually <span class="math-container">$sf(k(t)) = sY = S =I$</span>, so it is through increased investment. The <span class="math-container">$n$</span> and <span class="math-container">$g$</span> would be population and technological growth respectively and <span class="math-container">$\delta$</span> the rate of depreciation (although these are not important for intuition about raising level of capital).</p>
<p><strong>Does this push capital beyond steady state?</strong></p>
<p>Now this is surprisingly common misconception. Increase in savings rate can increase the level of capital stock - it just cannot increase the growth rate at which new capital per effective worker accumulates.</p>
<p>According to Romer Advanced Macroeconomics [emphasis mine]:</p>
<blockquote>
<p>, a <strong>change in the saving rate has a level effect but not a growth
effect</strong>: it changes the economy’s balanced growth path, and thus the level of
output per worker at any point in time, but it does not affect the growth
rate of output per worker on the balanced growth path.</p>
</blockquote>
<p>However, this is quite common misconception because of the confusing terminology. Increase in level in macroeconomic. Consider for example the visualization from Romer Advanced Macroeconomics pp 20:</p>
<p><a href="https://i.sstatic.net/jGjTZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jGjTZ.png" alt="enter image description here" /></a></p>
<p>The upper graph shows you the growth rate of capital per effective worker. Indeed here savings rate has only one-of impact that quickly dies out (in fact this is just level increase in disguise as growth).</p>
<p>However, on the bottom panel you can clearly see that the <strong>level</strong> of capital per effective worker is now higher thanks to higher savings rate.</p>

CC BY-SA 4.0
2020-11-13T18:36:46.927Z
2020-11-13T18:36:46.927Z
1muflon1
<p>Savings in macro are equal to investment <span class="math-container">$(S=I)$</span>. Investment is how you get new capital stock. When people save more they also by definition invest more and when they invest more there is more capital.</p>

CC BY-SA 4.0
2020-11-13T17:50:24.073Z
Thanks for the response, doesn't that new investment just push the capital per worker beyond the steady state? more simply, how do they invest more without increasing the capital per worker ratio?
user3280937
CC BY-SA 4.0
2020-11-13T17:50:45.330Z
$S$ doesn't have to equal $I$. An increase in savings by one household need not result in a firm buying more capital goods.  $S=I$ is an equilibrium outcome, not an accounting identity.
BKay
CC BY-SA 4.0
2020-11-13T17:59:16.440Z
@BKay actually assuming G=0 it would come up from identity. By definition private saving is income minus consumption Y-C. Now from national identity if we set G=0  then by definition Y=C+I now solve for investment I=Y-C so I=S. All derived just from identities so not really an equilibrium outcome
1muflon1
CC BY-SA 4.0
2020-11-13T18:07:42.093Z
If in the solow model the y-axis is investment (sY) why does an increase in investment shift the line and not just move across it?
user3280937
CC BY-SA 4.0
2020-11-14T10:06:08.557Z
@user3280937 I don't get what you are asking. Why would investment being on y-axis prevent the shift?
csilvia
CC BY-SA 4.0
2020-11-13T17:33:10.483Z
2020-11-13T17:33:10.483Z
csilvia
<p>I understand mathematically how it works. But what is the actual process intuitively.</p>
<p>Thanks</p>

<macroeconomics><saving><solow>
Why does a rise in the savings rate result in an increase in the capital stock intuitively?
user3280937
# Why does a rise in the savings rate result in an increase in the capital stock intuitively?

(Asked by: username_0 on 2020-11-13)

I understand mathematically how it works. But what is the actual process intuitively.

Thanks

**Tags:** macroeconomics, saving, solow


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## 2 answers

### Answer by username_1 on 2020-11-13. Score: 3

The story in the other answer is not fundamentally wrong but incomplete and bit inaccurate.

Saving does actually affect capital stock through investment in Solow model (assuming based on the Solow tag that is the model you want intuition for), but there is more complexity to it.

**What is saving**

Investment is actually equal not just to private saving but both private and public saving since saving is actually defined as (following Blanchard et al Macroeconomics: a European Perspective pp 51):

$$S \equiv Y - T-C$$

Combining this definition with output identity for closed economy $Y\equiv C+I+G$ gives us:

$$S = I +T -G$$

However, it is often convenient in Solow model to just ignore government by assuming $G=T$, in which case $S+I$.

**Why does saving rate increase capital stock:**

Next the way how savings rate affects capital accumulation is depicted in the picture I took from Romer Advanced Macroeconomics 4th ed pp 19.

[![enter image description here](https://i.sstatic.net/GDhM8.png)](https://i.sstatic.net/GDhM8.png)

As you can see the intuition really works this way. Here $s$ is the savings rate and $f(k)=Y$. Higher $s$ means that savings increases, for example if $Y=100$ with $s=0.5$, $S=50$ and with $s=0.7$, $S=70$. Then if we assume away government indeed $S=I$ so $I=70$.

This is why the evolution of capital accumulation (per unit of effective worker) is given by:

$$\dot{k}(t) = sf (k(t)) − (n + g +\delta)k(t)$$

where actually $sf(k(t)) = sY = S =I$, so it is through increased investment. The $n$ and $g$ would be population and technological growth respectively and $\delta$ the rate of depreciation (although these are not important for intuition about raising level of capital).

**Does this push capital beyond steady state?**

Now this is surprisingly common misconception. Increase in savings rate can increase the level of capital stock - it just cannot increase the growth rate at which new capital per effective worker accumulates.

According to Romer Advanced Macroeconomics [emphasis mine]:

> , a **change in the saving rate has a level effect but not a growth
> effect**: it changes the economy’s balanced growth path, and thus the level of
> output per worker at any point in time, but it does not affect the growth
> rate of output per worker on the balanced growth path.

However, this is quite common misconception because of the confusing terminology. Increase in level in macroeconomic. Consider for example the visualization from Romer Advanced Macroeconomics pp 20:

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

The upper graph shows you the growth rate of capital per effective worker. Indeed here savings rate has only one-of impact that quickly dies out (in fact this is just level increase in disguise as growth).

However, on the bottom panel you can clearly see that the **level** of capital per effective worker is now higher thanks to higher savings rate.


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### Answer by username_2 on 2020-11-13. Score: 2

Savings in macro are equal to investment $(S=I)$. Investment is how you get new capital stock. When people save more they also by definition invest more and when they invest more there is more capital.

> **Comments:**
> **username_0**: Thanks for the response, doesn't that new investment just push the capital per worker beyond the steady state? more simply, how do they invest more without increasing the capital per worker ratio?  
> **username_3**: $S$ doesn't have to equal $I$. An increase in savings by one household need not result in a firm buying more capital goods.  $S=I$ is an equilibrium outcome, not an accounting identity.  
> **username_1**: @username_3 actually assuming G=0 it would come up from identity. By definition private saving is income minus consumption Y-C. Now from national identity if we set G=0  then by definition Y=C+I now solve for investment I=Y-C so I=S. All derived just from identities so not really an equilibrium outcome  
> **username_0**: If in the solow model the y-axis is investment (sY) why does an increase in investment shift the line and not just move across it?  
> **username_2**: @username_0 I don't get what you are asking. Why would investment being on y-axis prevent the shift?  

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