<p><strong>tl;dr</strong></p>
<p>There is absolutely no contradiction. <span class="math-container">$I=S$</span> is purely definitional like saying all triangles have three angles. It holds always because we defined the word saving in such way to always be equal to investment same way as we defined word triangle as to  always refer to object with 3 angles.</p>
<p>You seem to be committing equivocation fallacy by defining saving and investment in a layman terms (average joe on a street does not use saving and investment in their rigorous scientific definitions), instead of their proper economic definitions, and then wondering why economic models that use rigorous definitions do not seem to fit. The fallacy would be equivalent to wondering how come that in physics moon is satellite when (in layman's understanding) satellite is some man-made object in space and well people did not build moon.</p>
<p>If you use proper definitions there are no contradictions. Increase in interest rate actually has both negative and positive effect on savings  (as it has both effect that lower and increase income and saving is properly defined as difference between net income and consumption). Due to way how saving is defined even if people put less money into financial assets (deposits at bank etc ) there is not necessarily less saving in the economy.</p>
<p><strong>Full Answer:</strong></p>
<blockquote>
<p>It seems to me that this &quot;S must equal I&quot; is contradicted by the logic behind monetary policy. Interest rate policy seems to based on savings can be put aside somewhere and nothing is done with them. BUT S=I is the mantra of every economics textbook/model.</p>
</blockquote>
<p>It is not contradicted by monetary policy.</p>
<p>First, you seem not to understand what <span class="math-container">$I=S$</span> means or how <span class="math-container">$I$</span> or <span class="math-container">$S$</span> is even defined in economics so we have to cover that first.  <span class="math-container">$I=S$</span> <strong>is not mantra</strong> it is <strong>definition</strong>.  That would be like saying in physics saying that shape of earth is oblate spheroid is a mantra, or saying that in geometry a shape that has 3 angles is called triangle is a mantra. Those are all definitions.</p>
<p>Likewise, <span class="math-container">$I=S$</span> is not some empirical observation it is consequence of us defining investment and saving in such way by construction. If you define word 'triangle' to mean any shape that has three angles then its not some sort of arcane mystery or magic that anything with three angles will be labeled triangle.</p>
<p>Furthermore, this definition is only valid when government runs balanced budget in closed economy.  The <span class="math-container">$I=S$</span> comes from definition of GDP in closed economy when government runs balanced budget. By <strong>definition</strong> GDP in closed economy is given by:</p>
<p><span class="math-container">$$Y = C+ I +G$$</span></p>
<p>Where <span class="math-container">$Y$</span> is output/income, <span class="math-container">$C$</span> private consumption, <span class="math-container">$I$</span> investment and <span class="math-container">$G$</span> public consumption. <strong>By definition</strong> saving is <span class="math-container">$S=(Y-T)-C$</span>, where <span class="math-container">$T$</span> are taxes, hence saving is not simply putting money in bank. Any time you do not devote your income to consumption of final goods and services you are saving under the definition above.  What even more any time government taxes you, the economy is actually increasing <strong>public savings</strong>. Using both definition of <span class="math-container">$Y$</span> and <span class="math-container">$S$</span> (by substituting the definition of <span class="math-container">$S$</span> into definition of <span class="math-container">$Y$</span> by solving for <span class="math-container">$Y$</span>) we arrive at:</p>
<p><span class="math-container">$$S + T + C = C + I +G \implies S +G -T = I $$</span></p>
<p>Now if we assume <strong>balanced budget</strong> and hence no <strong>public saving</strong> we will have <span class="math-container">$G=T \implies S=I$</span>. This purely definitional like saying that triangles have three angles, that degrees in circle sum up to 360 degrees etc. It is not empirical observation, its just a definitional. Note the terms are again not even defined as regular people understand them. Investment for example includes inventory investment. Hence any time firm is not selling some goods that would be considered investing. That is not the layman person definition of investment but all sciences have specialized language (e.g. layman would not call moon satellite yet that's astrophysics definition).</p>
<p>Hence we have:</p>
<p><span class="math-container">$S = I$</span> or in words investment equals private saving, in closed economy with balanced budget.</p>
<p><span class="math-container">$S + T-G= I $</span> or investment equals private plus public saving, in closed economy without balanced budget.</p>
<p><span class="math-container">$S + T-G +NX = I $</span> or saving equals private plus public plus saving by accumulation of trade surplus, in open economy without balanced budget.</p>
<p>We always have different kinds of saving on LHS and investment on RHS so some textbooks will just state saving equals investment, however that is not the same as <span class="math-container">$S=I$</span> which would be only <strong>private saving</strong> equals investment.</p>
<p>Second, you are not completely correct in saying:</p>
<blockquote>
<p>But when it comes to monetary policy, central banks increase interest rates in order to slow inflation (and in doing so slow investment). But higher interest rates will increase the attractiveness of saving thus increasing investment (S=I)</p>
</blockquote>
<p>A basic textbook framework for understanding monetary policy is IS-LM model (I will use the closed economy version to save time). In IS-LM model we have the IS (interest saving relation):</p>
<p><span class="math-container">$$Y = C(Y-T) + I(Y,i) + G$$</span></p>
<p>Which is the same formula as definition for GDP just with emphasis that <span class="math-container">$C$</span>, and <span class="math-container">$I$</span> is also a function of other parameters like <span class="math-container">$Y-T$</span>, <span class="math-container">$Y$</span> or <span class="math-container">$i$</span>. The whole reason why this is called interest saving relation is that <span class="math-container">$Y = C(Y-T) + I(Y,i) + G \implies S +G -T = I$</span>, as already proved above. Again this is just definitional like that triangle have 3 angles or that shape ball has is sphere etc. This is not fundamental truth or empirical observation, it is just a consequence of people who first came up with these definitions like Keynes and others, defining saving, investment etc in such ways.</p>
<p>Next we have the LM (liquidity-money) relation given by:</p>
<p><span class="math-container">$$M/P = L(Y,i)$$</span></p>
<p>where <span class="math-container">$M$</span> is money supply, <span class="math-container">$P$</span> price level (increase in which gives you inflation) and <span class="math-container">$L$</span> which is money demand that depends on <span class="math-container">$Y$</span> and <span class="math-container">$i$</span>.</p>
<p>Now to further solve the model we have to concretely define the functional form of money demand <span class="math-container">$L(Y,i)$</span> consumption <span class="math-container">$C(Y-T)$</span>, and investment <span class="math-container">$I(Y,i)$</span>, for the sake of simplicity I will use basic linear functions for all of them, the exact functional form does not matter but mathematically linear functions are easy to solve. Hence let's say consumption is given by:</p>
<p><span class="math-container">$$C = c_0 + c_1(Y-T)$$</span></p>
<p>Where <span class="math-container">$c_0$</span> is autonomous consumption and <span class="math-container">$c_1$</span> marginal propensity to consume.</p>
<p>Investment is given by:</p>
<p><span class="math-container">$$I  = I_0 + d_1 Y - d_2 i$$</span></p>
<p>where <span class="math-container">$I_0$</span> is autonomous investment and <span class="math-container">$d_1$</span> and <span class="math-container">$d_2$</span> are effects of income and interest rate.</p>
<p>Furthermore, note this also implies that <span class="math-container">$S +  T -G=   I_0 + d_1 Y - d_2 i$</span> so it is not true that low interest rate discourages saving <strong>directly</strong>. However, note here saving does not mean putting money inside your bank deposit, saving is difference between income and consumption, lower investment lowers income so it has to lower also saving because you have less income from which you can save money.</p>
<p>Higher interest rate does encourage people to put money inside bank or other financial assets as opposed to saving by not consuming your resources in form of improving your business  or other forms of not consuming your resources (which is how saving is actually defined). For example, higher interest rates would affect negatively your saving in form of forming inventory etc. Now as we will see later lower interest rate also encourage economic activity by stimulating <span class="math-container">$Y$</span> and that also increases saving (as defined in macro), so there are both positive and negative effects of <span class="math-container">$i$</span> on saving and simply stating that <span class="math-container">$i$</span> encourages saving, as defined in macro, is not correct (not in general equilibrium model anyway, sure if you just have partial equilibrium of financial market (e.g. market for bank deposits) then higher interest rate means more supply of saving, but that's only one form of saving not all saving in the macroeconomy.</p>
<p>Next, the money demand can be defined as:</p>
<p><span class="math-container">$$L  = f_1 Y - f_2 i$$</span></p>
<p>using the definitions above we have:</p>
<p><span class="math-container">$$\text{IS: } \quad Y = c_0 + c_1(Y-T) + I_0 + d_1 Y - d_2 i + G \implies Y = \frac{1}{1-c_1-d_1} \left( c_0 + I_0 +G -c_1T \right)- \frac{1}{1-c_1-d_2} i $$</span></p>
<p><span class="math-container">$$\text{LM:  } \quad M/P =  f_1 Y - f_2 i \implies Y= \frac{1}{f_1} \frac{M}{P} + \frac{f_2}{f_1} i$$</span></p>
<p>So as we can see above interest rate has both negative and positive effect on output and thus also on level of saving. Changes in interest rates certainly can change the relative share of <span class="math-container">$S$</span>, <span class="math-container">$G$</span>, <span class="math-container">$T$</span> or <span class="math-container">$I$</span> in the economy but the identity <span class="math-container">$S +G -T = I$</span> will still perfectly hold. Moreover, if we subdivide <span class="math-container">$S$</span> into saving in form of deposits <span class="math-container">$S_1$</span>, saving in form of accumulating more inventory <span class="math-container">$S_2$</span> etc, we could again see changes in interest rate changing composition of <span class="math-container">$S$</span> between <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span>, but the overall identity <span class="math-container">$S+G-T = I$</span> is left unchanged. If we assume that government runs balanced budget then we can make the same claims about <span class="math-container">$S=I$</span>. Again, interest rate increase may encourage saving in some form (e.g. deposits) but it discourages saving in other forms (e.g. accumulating inventory, putting money under mattress).</p>

CC BY-SA 4.0
2022-05-11T12:27:00.860Z
@user253751 taxes
1muflon1
CC BY-SA 4.0
2022-05-11T17:09:42.083Z
@user253751 because S is private saving people just sometimes abbreviate private saving to just saving but S only looks at private saving, government is not private entity also government saving is by definition T-G
1muflon1
CC BY-SA 4.0
2022-06-13T06:39:32.037Z
Thanks for that wonderful answer. I see what you are saying. So what should we call the savings and investment that are affected by interest rates? The two things that move counter to each other and we call savings and investments in the media, politicians and pretty much everyone else outside economics?
Studi
CC BY-SA 4.0
2022-06-13T06:47:46.057Z
@Food in economics they don’t have special name. Also on micro level all types of saving and investment respond differently. Bonds don’t respond equally as stocks etc
1muflon1
CC BY-SA 4.0
2022-06-13T06:59:46.217Z
Thanks muflon. I appreciate it. I have posted another question about this. I hope it doesn't cause too much ire!
Studi
CC BY-SA 4.0
2022-05-11T10:34:42.833Z
2022-05-11T12:31:28.247Z
2022-05-11T12:31:28.247Z
1muflon1
<p>If savings = investment, then by definition, higher interest rates encourage savings and hence more investment. But when it comes to monetary policy,  central banks increase interest rates in order to slow inflation (and in doing so slow investment). But higher interest rates will increase the attractiveness of saving thus increasing investment (S=I) .</p>
<p>It seems to me that this &quot;S must equal I&quot; is contradicted by the logic behind monetary policy. Interest rate policy seems to based on  savings can be put aside somewhere and nothing is done with them. BUT S=I is the mantra of every economics textbook/model.</p>
<p>I am going in circles here!</p>

CC BY-SA 4.0
2022-05-11T08:39:04.413Z
$S=I$ is a macroeconomic tautology in a closed economy. It is not really about money, but instead about stuff, with $S$ being the difference between production and non-capital consumption, and $I$ being how that difference is used locally. It is not true in an open economy with international trade and financial flows. Changing interest rates affect the relative attractiveness of consumption and investment, and pushes towards a different overall equilibrium for the economy.
Henry
<macroeconomics><inflation><interest-rate><monetary-policy><central-banking>
Savings = Investment. High interest rates stifle investment. High interest rate encourage savings. Isn't this a contradiction?
Studi
# Savings = Investment. High interest rates stifle investment. High interest rate encourage savings. Isn't this a contradiction?

(Asked by: username_0 on 2022-05-11)

If savings = investment, then by definition, higher interest rates encourage savings and hence more investment. But when it comes to monetary policy, central banks increase interest rates in order to slow inflation (and in doing so slow investment). But higher interest rates will increase the attractiveness of saving thus increasing investment (S=I) .

It seems to me that this "S must equal I" is contradicted by the logic behind monetary policy. Interest rate policy seems to based on savings can be put aside somewhere and nothing is done with them. BUT S=I is the mantra of every economics textbook/model.

I am going in circles here!

**Tags:** macroeconomics, inflation, interest-rate, monetary-policy, central-banking
> **Comments:**
> **username_1**: $S=I$ is a macroeconomic tautology in a closed economy. It is not really about money, but instead about stuff, with $S$ being the difference between production and non-capital consumption, and $I$ being how that difference is used locally. It is not true in an open economy with international trade and financial flows. Changing interest rates affect the relative attractiveness of consumption and investment, and pushes towards a different overall equilibrium for the economy.  

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## 1 answers

### Answer by username_2 on 2022-05-11. Score: 2

**tl;dr**

There is absolutely no contradiction. $I=S$ is purely definitional like saying all triangles have three angles. It holds always because we defined the word saving in such way to always be equal to investment same way as we defined word triangle as to always refer to object with 3 angles.

You seem to be committing equivocation fallacy by defining saving and investment in a layman terms (average joe on a street does not use saving and investment in their rigorous scientific definitions), instead of their proper economic definitions, and then wondering why economic models that use rigorous definitions do not seem to fit. The fallacy would be equivalent to wondering how come that in physics moon is satellite when (in layman's understanding) satellite is some man-made object in space and well people did not build moon.

If you use proper definitions there are no contradictions. Increase in interest rate actually has both negative and positive effect on savings (as it has both effect that lower and increase income and saving is properly defined as difference between net income and consumption). Due to way how saving is defined even if people put less money into financial assets (deposits at bank etc ) there is not necessarily less saving in the economy.

**Full Answer:**

> It seems to me that this "S must equal I" is contradicted by the logic behind monetary policy. Interest rate policy seems to based on savings can be put aside somewhere and nothing is done with them. BUT S=I is the mantra of every economics textbook/model.

It is not contradicted by monetary policy.

First, you seem not to understand what $I=S$ means or how $I$ or $S$ is even defined in economics so we have to cover that first. $I=S$ **is not mantra** it is **definition**. That would be like saying in physics saying that shape of earth is oblate spheroid is a mantra, or saying that in geometry a shape that has 3 angles is called triangle is a mantra. Those are all definitions.

Likewise, $I=S$ is not some empirical observation it is consequence of us defining investment and saving in such way by construction. If you define word 'triangle' to mean any shape that has three angles then its not some sort of arcane mystery or magic that anything with three angles will be labeled triangle.

Furthermore, this definition is only valid when government runs balanced budget in closed economy. The $I=S$ comes from definition of GDP in closed economy when government runs balanced budget. By **definition** GDP in closed economy is given by:

$$Y = C+ I +G$$

Where $Y$ is output/income, $C$ private consumption, $I$ investment and $G$ public consumption. **By definition** saving is $S=(Y-T)-C$, where $T$ are taxes, hence saving is not simply putting money in bank. Any time you do not devote your income to consumption of final goods and services you are saving under the definition above. What even more any time government taxes you, the economy is actually increasing **public savings**. Using both definition of $Y$ and $S$ (by substituting the definition of $S$ into definition of $Y$ by solving for $Y$) we arrive at:

$$S + T + C = C + I +G \implies S +G -T = I $$

Now if we assume **balanced budget** and hence no **public saving** we will have $G=T \implies S=I$. This purely definitional like saying that triangles have three angles, that degrees in circle sum up to 360 degrees etc. It is not empirical observation, its just a definitional. Note the terms are again not even defined as regular people understand them. Investment for example includes inventory investment. Hence any time firm is not selling some goods that would be considered investing. That is not the layman person definition of investment but all sciences have specialized language (e.g. layman would not call moon satellite yet that's astrophysics definition).

Hence we have:

$S = I$ or in words investment equals private saving, in closed economy with balanced budget.

$S + T-G= I $ or investment equals private plus public saving, in closed economy without balanced budget.

$S + T-G +NX = I $ or saving equals private plus public plus saving by accumulation of trade surplus, in open economy without balanced budget.

We always have different kinds of saving on LHS and investment on RHS so some textbooks will just state saving equals investment, however that is not the same as $S=I$ which would be only **private saving** equals investment.

Second, you are not completely correct in saying:

> But when it comes to monetary policy, central banks increase interest rates in order to slow inflation (and in doing so slow investment). But higher interest rates will increase the attractiveness of saving thus increasing investment (S=I)

A basic textbook framework for understanding monetary policy is IS-LM model (I will use the closed economy version to save time). In IS-LM model we have the IS (interest saving relation):

$$Y = C(Y-T) + I(Y,i) + G$$

Which is the same formula as definition for GDP just with emphasis that $C$, and $I$ is also a function of other parameters like $Y-T$, $Y$ or $i$. The whole reason why this is called interest saving relation is that $Y = C(Y-T) + I(Y,i) + G \implies S +G -T = I$, as already proved above. Again this is just definitional like that triangle have 3 angles or that shape ball has is sphere etc. This is not fundamental truth or empirical observation, it is just a consequence of people who first came up with these definitions like Keynes and others, defining saving, investment etc in such ways.

Next we have the LM (liquidity-money) relation given by:

$$M/P = L(Y,i)$$

where $M$ is money supply, $P$ price level (increase in which gives you inflation) and $L$ which is money demand that depends on $Y$ and $i$.

Now to further solve the model we have to concretely define the functional form of money demand $L(Y,i)$ consumption $C(Y-T)$, and investment $I(Y,i)$, for the sake of simplicity I will use basic linear functions for all of them, the exact functional form does not matter but mathematically linear functions are easy to solve. Hence let's say consumption is given by:

$$C = c\_0 + c\_1(Y-T)$$

Where $c\_0$ is autonomous consumption and $c\_1$ marginal propensity to consume.

Investment is given by:

$$I = I\_0 + d\_1 Y - d\_2 i$$

where $I\_0$ is autonomous investment and $d\_1$ and $d\_2$ are effects of income and interest rate.

Furthermore, note this also implies that $S + T -G= I\_0 + d\_1 Y - d\_2 i$ so it is not true that low interest rate discourages saving **directly**. However, note here saving does not mean putting money inside your bank deposit, saving is difference between income and consumption, lower investment lowers income so it has to lower also saving because you have less income from which you can save money.

Higher interest rate does encourage people to put money inside bank or other financial assets as opposed to saving by not consuming your resources in form of improving your business or other forms of not consuming your resources (which is how saving is actually defined). For example, higher interest rates would affect negatively your saving in form of forming inventory etc. Now as we will see later lower interest rate also encourage economic activity by stimulating $Y$ and that also increases saving (as defined in macro), so there are both positive and negative effects of $i$ on saving and simply stating that $i$ encourages saving, as defined in macro, is not correct (not in general equilibrium model anyway, sure if you just have partial equilibrium of financial market (e.g. market for bank deposits) then higher interest rate means more supply of saving, but that's only one form of saving not all saving in the macroeconomy.

Next, the money demand can be defined as:

$$L = f\_1 Y - f\_2 i$$

using the definitions above we have:

$$\text{IS: } \quad Y = c\_0 + c\_1(Y-T) + I\_0 + d\_1 Y - d\_2 i + G \implies Y = \frac{1}{1-c\_1-d\_1} \left( c\_0 + I\_0 +G -c\_1T \right)- \frac{1}{1-c\_1-d\_2} i $$

$$\text{LM: } \quad M/P = f\_1 Y - f\_2 i \implies Y= \frac{1}{f\_1} \frac{M}{P} + \frac{f\_2}{f\_1} i$$

So as we can see above interest rate has both negative and positive effect on output and thus also on level of saving. Changes in interest rates certainly can change the relative share of $S$, $G$, $T$ or $I$ in the economy but the identity $S +G -T = I$ will still perfectly hold. Moreover, if we subdivide $S$ into saving in form of deposits $S\_1$, saving in form of accumulating more inventory $S\_2$ etc, we could again see changes in interest rate changing composition of $S$ between $S\_1$ and $S\_2$, but the overall identity $S+G-T = I$ is left unchanged. If we assume that government runs balanced budget then we can make the same claims about $S=I$. Again, interest rate increase may encourage saving in some form (e.g. deposits) but it discourages saving in other forms (e.g. accumulating inventory, putting money under mattress).

> **Comments:**
> **username_2**: @user253751 taxes  
> **username_2**: @user253751 because S is private saving people just sometimes abbreviate private saving to just saving but S only looks at private saving, government is not private entity also government saving is by definition T-G  
> **username_0**: Thanks for that wonderful answer. I see what you are saying. So what should we call the savings and investment that are affected by interest rates? The two things that move counter to each other and we call savings and investments in the media, politicians and pretty much everyone else outside economics?  
> **username_2**: @Food in economics they don’t have special name. Also on micro level all types of saving and investment respond differently. Bonds don’t respond equally as stocks etc  
> **username_0**: Thanks muflon. I appreciate it. I have posted another question about this. I hope it doesn't cause too much ire!  

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