<p>Kenny LJ and FooBar already did a great job of clarifying the broader conceptual issue here: in equilibrium, prices don't just increase or decrease without a cause (a shift in supply or demand), and any deviation from the equilibrium price will result in a surplus or deficit.</p>

<p>But since you're talking about interest rates, I assume that you're interested in what happens when a central bank changes the interest rate. This is legitimately <em>very</em> confusing, because at first glance it doesn't <em>seem</em> like the central bank is affecting either the supply of savings or the demand for investment - and hence, by changing the interest rate, it <em>seems</em> like the central bank will break the equality of savings and investment. </p>

<p>This isn't true. But it took me years to wrap my head around this, and I suspect that the vast majority of economists would get it wrong if asked to explain it (as evinced, perhaps, that someone as smart as FooBar misfired a little on this point). Indeed, John Taylor - an extremely well-known macroeconomist - recently fell victim to a fallacy here, as Miles Kimball pointed out (<a href="http://blog.supplysideliberal.com/post/41785783895/contra-john-taylor" rel="nofollow">see point 7 here</a>; as further proof of the difficulty of this topic, I'm not sure that Kimball got it quite right either!). </p>

<p>To clarify, let's split into two cases.</p>

<p><strong>Case 1: flexible prices.</strong></p>

<p>Note that what really matters for savings and investment is the <em>real</em> interest rate, which is the nominal interest rate minus expected inflation, while the central bank sets the <em>nominal</em> interest rate. Suppose that the central bank increases interest rates, and at first suppose that inflation expectations remain constant. Then as you point out, desired savings will increase while desired investment falls. </p>

<p>This means that there will be a glut in the economy: there will be more goods supplied than demanded. How is this resolved? In a world where prices are flexible, the price will fall until inflation expectations <em>rise</em> exactly enough to offset the central bank's increase in the nominal interest rate, leaving the real interest rate ultimately unchanged (so that desired savings and investment balance again). For instance, if we have a certain expectation of where prices will be next year, then when prices fall today, that fall creates more expected inflation over the next year.</p>

<p>This is the neoclassical, flexible price case, and it resolves your paradox in the simplest possible way - by saying that expected inflation will adjust such that real interest rates, and hence the desire to save and invest, are unchanged. But as you can probably guess, this is fairly unrealistic: prices don't always adjust that quickly, and it's counterfactual to say that higher nominal interest rates result in an equal rise in expected inflation.</p>

<p><strong>Case 2: sticky prices.</strong></p>

<p>Let's suppose that prices are "sticky", meaning that they don't change in response to events very quickly. This is the kind of assumption made in "New Keynesian" models (and, to some extent, "Old Keynesian" and monetarist models before them). </p>

<p>Indeed, for our purposes, let's suppose that prices are <em>so</em> sticky that expected inflation is a constant, so that a change in the nominal interest rate translates exactly one-for-one to a change in the real interest rate. (This is not a bad assumption empirically, <a href="http://www.columbia.edu/~en2198/papers/realrate.pdf" rel="nofollow">according to Nakamura and Steinsson</a>!) This means that we can't use the dodge from before: when the central bank pushes up the nominal interest rate, the real interest rate that matters for the savings/investment market rises too, and it <em>seems</em> like we should get some kind of disequilibrium.</p>

<p>Indeed, if not all sides of the market react right away, something like this might happen. Suppose that in response to an interest rate hike, consumers quickly increase their savings, cutting back consumption, but producers are slow to respond and keep producing the same amount of goods. (Let's ignore the response of desired investment to interest rates, which complicates the story but doesn't change the basic lessons.)</p>

<p>Then there will be a surplus of unsold goods, which will accumulate as <em>inventories</em>. This inventory accumulation is a form of investment (e.g. <a href="http://bea.gov/iTable/iTableHtml.cfm?reqid=9&amp;step=3&amp;isuri=1&amp;903=5" rel="nofollow">line 14 in the US national accounts here</a>), so $S=I$ will continue to hold - but it will hold in a <em>weird</em> way, where consumers' saving is matched by producers' unintended inventory accumulation.</p>

<p>Alternatively, suppose that producers react right away. Then when consumers try to save more and consume less, producers will produce less, and there will be less income earned by workers and capitalists. How will consumers deal with this fall in income? Partly, they'll save less - because it makes less sense to save when your earnings are low relative to their usual level. Partly, though, they'll consume even less - which means that producers make less and incomes fall again, and so on and so on. </p>

<p>This is just the <a href="http://en.wikipedia.org/wiki/Multiplier_%28economics%29" rel="nofollow">Keynesian multiplier</a>, and the process ends when income $Y$ has fallen by $\Delta Y = \Delta S/MPS$, where $\Delta S$ is the initial increase in desired savings and $MPS$ is the marginal propensity to save out of income. At the point, the decrease $MPS\times \Delta Y$ in desired savings from the $\Delta Y$ fall in income precisely offsets the initial increase $\Delta S$ in desired savings arising from the higher interest rate. (If there is an initial fall $\Delta I$ in investment in response to the higher interest rate too, then this becomes $\Delta Y = (\Delta S - \Delta I)/MPS$; and it becomes more complicated still if investment changes in response to income, etc..)</p>

<p>In short, $S=I$ continues to hold because <em>desired savings depends on income</em> - which you can't see from the simple, partial equilibrium diagram showing S and I relative to the interest rate. Consumption (and therefore income) will eventually fall enough to cause a decline in savings that restores equilibrium.</p>

CC BY-SA 3.0
2015-05-09T02:48:31.370Z
2015-05-09T02:48:31.370Z
nominally rigid
<h3>Are prices exogenous?</h3>

<p>You assumed an exogenous shift in the interest rate. Typically, prices are endogenous in Macroeconomics (and the interest rate is nothing but the price of capital). Absent frictions, prices adjust such that supply equals demand. </p>

<p>That is exactly why your thought experiment doesn't make much sense in the very stylized model: Suppose we somehow raise the interest rate, then supply does not equal demand anymore (unless there is storage, see below). Hence, the <a href="http://en.wikipedia.org/wiki/Walrasian_auction" rel="nofollow">Walrasian auctioneer</a> will decrease the interest rate until supply equals demand again - the original point.</p>

<h3>Can there be exogenous shocks to prices?</h3>

<p>In a more sophisticated world, exogenous factors such as monetary policy affect the interest rate. Why? Well, because actually savings needn't be investment. For example, if households just hold cash at home and do not spend it, that is savings (defined as income minus consumption), but it is not invested. Similar stories hold for banks that hold cash (or store their money at their central bank).</p>

CC BY-SA 3.0
2015-05-07T19:25:46.247Z
Consider the following argument: the rise in interest rate caused business to borrow less for investment, and consequently the production of the capital will decrease, which consequently cause the national income to decrease. So people does not have as much income as before when the interest rate has not rose, so their saving is clearly also decreased.        Do you think it is an valid argument?
Kun
CC BY-SA 3.0
2015-05-07T19:28:19.347Z
@Kund that depends on your timing assumption. Typically, investment *now* yields additional production *in the future* - there is some delay. Hence, a decrease in investment today will not decrease income today, but rather a decrease in income tomorrow.
FooBar
CC BY-SA 3.0
2015-05-07T19:36:05.393Z
I mean the investment money now would yields production of capital today. I mean the production of capital instead of what can be produced by the capital in the future.
Kun
CC BY-SA 3.0
2015-05-09T02:55:57.480Z
@FooBar, the 1st half of your answer is great, but I have to strongly disagree with the 2nd half. In a closed economy, when savings and investment are defined properly (combining public and private sectors), S=I will always hold. Your example of households and cash isn't right: if households in the aggregate demand more cash, then either the central bank will trade them that cash for other assets (in which case private & public savings are both unchanged), or government will issue more liabilities (in which case private savings rise and public savings fall, with no net _aggregate_ change).
nominally rigid
CC BY-SA 3.0
2015-05-09T02:58:10.450Z
@Kun, yes, I think your argument has the right idea, and I make a more elaborate version of this argument in my answer. When a rise in interest rates causes less desired investment and more desired savings (meaning less desired consumption), income will fall and keep falling until savings decline enough to restore equilibrium.
nominally rigid
CC BY-SA 3.0
2015-05-09T10:57:20.037Z
@nominallyrigid I'll might answer more as I'm in a hurry, but two points (i) I should've mentioned sticky prices. My answer was completely out of NK context or monetary policy: I see the issue with cash holdings, but they should've just been an example for storage devices. Now, for example, if inventories (which are technically storage device) are *defined* to be investment, and if similar storage devices all are defined as such, then *when savings and investment are defined properly, S=I will always hold* might be tautologically true.
FooBar
CC BY-SA 3.0
2015-05-09T17:13:23.030Z
@FooBar, S=I is in one sense a tautology, but it's a tautology in the same sense that income=production or eq supply=eq demand are, and like them it has pretty profound consequences. Note the national accounts are set up to enforce S=I, albeit adjusted for the open economy: net savings equal net investment plus the current account (up to the GDP/GDI statistical discrepancy); see [here](http://research.stlouisfed.org/fred2/graph/?g=1aOU).
nominally rigid
CC BY-SA 3.0
2015-05-07T18:48:56.257Z
2015-05-07T18:48:56.257Z
FooBar
<p>It is stated that saving equals investment. However, suppose the interest rate rises and people save more. However, businesses will borrow less so investment decreases. Hence investment is less than saving now. What happened?</p>

CC BY-SA 3.0
2015-05-07T17:45:07.960Z
Are you familiar with "IS-LM" at all? If not, that's probably the best google query for you to start learning more. My question to you is how do you know that the rise in interest rates wasn't from a "shift in investment" in the sense that at any interest rate firms are now investing relatively more than beforehand? People would save more, the interest rate would be higher, and investment would also be higher.
Hessian
CC BY-SA 3.0
2015-05-07T17:48:52.300Z
I guess I will start to understand it in the intermediate economics text.
Kun
CC BY-SA 3.0
2015-05-09T02:49:32.703Z
I disagree with the close votes here. It might not be perfectly stated, but (presuming that the interest rate rise is coming from the central bank) this is a very deep and difficult question that most economists would struggle to answer.
nominally rigid
<macroeconomics>
How do saving and investment work in macroeconomics?
Kun
# How do saving and investment work in macroeconomics?

(Asked by: username_0 on 2015-05-07)

It is stated that saving equals investment. However, suppose the interest rate rises and people save more. However, businesses will borrow less so investment decreases. Hence investment is less than saving now. What happened?

**Tags:** macroeconomics

> **Comments:**
> **username_1**: Are you familiar with "IS-LM" at all? If not, that's probably the best google query for you to start learning more. My question to you is how do you know that the rise in interest rates wasn't from a "shift in investment" in the sense that at any interest rate firms are now investing relatively more than beforehand? People would save more, the interest rate would be higher, and investment would also be higher.  
> **username_0**: I guess I will start to understand it in the intermediate economics text.  
> **username_2**: I disagree with the close votes here. It might not be perfectly stated, but (presuming that the interest rate rise is coming from the central bank) this is a very deep and difficult question that most economists would struggle to answer.  

---

## 2 answers

### Answer by username_2 on 2015-05-09. Score: 2

Kenny LJ and username_3 already did a great job of clarifying the broader conceptual issue here: in equilibrium, prices don't just increase or decrease without a cause (a shift in supply or demand), and any deviation from the equilibrium price will result in a surplus or deficit.

But since you're talking about interest rates, I assume that you're interested in what happens when a central bank changes the interest rate. This is legitimately *very* confusing, because at first glance it doesn't *seem* like the central bank is affecting either the supply of savings or the demand for investment - and hence, by changing the interest rate, it *seems* like the central bank will break the equality of savings and investment.

This isn't true. But it took me years to wrap my head around this, and I suspect that the vast majority of economists would get it wrong if asked to explain it (as evinced, perhaps, that someone as smart as username_3 misfired a little on this point). Indeed, John Taylor - an extremely well-known macroeconomist - recently fell victim to a fallacy here, as Miles Kimball pointed out ([see point 7 here](http://blog.supplysideliberal.com/post/41785783895/contra-john-taylor); as further proof of the difficulty of this topic, I'm not sure that Kimball got it quite right either!).

To clarify, let's split into two cases.

**Case 1: flexible prices.**

Note that what really matters for savings and investment is the *real* interest rate, which is the nominal interest rate minus expected inflation, while the central bank sets the *nominal* interest rate. Suppose that the central bank increases interest rates, and at first suppose that inflation expectations remain constant. Then as you point out, desired savings will increase while desired investment falls.

This means that there will be a glut in the economy: there will be more goods supplied than demanded. How is this resolved? In a world where prices are flexible, the price will fall until inflation expectations *rise* exactly enough to offset the central bank's increase in the nominal interest rate, leaving the real interest rate ultimately unchanged (so that desired savings and investment balance again). For instance, if we have a certain expectation of where prices will be next year, then when prices fall today, that fall creates more expected inflation over the next year.

This is the neoclassical, flexible price case, and it resolves your paradox in the simplest possible way - by saying that expected inflation will adjust such that real interest rates, and hence the desire to save and invest, are unchanged. But as you can probably guess, this is fairly unrealistic: prices don't always adjust that quickly, and it's counterfactual to say that higher nominal interest rates result in an equal rise in expected inflation.

**Case 2: sticky prices.**

Let's suppose that prices are "sticky", meaning that they don't change in response to events very quickly. This is the kind of assumption made in "New Keynesian" models (and, to some extent, "Old Keynesian" and monetarist models before them).

Indeed, for our purposes, let's suppose that prices are *so* sticky that expected inflation is a constant, so that a change in the nominal interest rate translates exactly one-for-one to a change in the real interest rate. (This is not a bad assumption empirically, [according to Nakamura and Steinsson](http://www.columbia.edu/~en2198/papers/realrate.pdf)!) This means that we can't use the dodge from before: when the central bank pushes up the nominal interest rate, the real interest rate that matters for the savings/investment market rises too, and it *seems* like we should get some kind of disequilibrium.

Indeed, if not all sides of the market react right away, something like this might happen. Suppose that in response to an interest rate hike, consumers quickly increase their savings, cutting back consumption, but producers are slow to respond and keep producing the same amount of goods. (Let's ignore the response of desired investment to interest rates, which complicates the story but doesn't change the basic lessons.)

Then there will be a surplus of unsold goods, which will accumulate as *inventories*. This inventory accumulation is a form of investment (e.g. [line 14 in the US national accounts here](http://bea.gov/iTable/iTableHtml.cfm?reqid=9&step=3&isuri=1&903=5)), so $S=I$ will continue to hold - but it will hold in a *weird* way, where consumers' saving is matched by producers' unintended inventory accumulation.

Alternatively, suppose that producers react right away. Then when consumers try to save more and consume less, producers will produce less, and there will be less income earned by workers and capitalists. How will consumers deal with this fall in income? Partly, they'll save less - because it makes less sense to save when your earnings are low relative to their usual level. Partly, though, they'll consume even less - which means that producers make less and incomes fall again, and so on and so on.

This is just the [Keynesian multiplier](http://en.wikipedia.org/wiki/Multiplier_%28economics%29), and the process ends when income $Y$ has fallen by $\Delta Y = \Delta S/MPS$, where $\Delta S$ is the initial increase in desired savings and $MPS$ is the marginal propensity to save out of income. At the point, the decrease $MPS\times \Delta Y$ in desired savings from the $\Delta Y$ fall in income precisely offsets the initial increase $\Delta S$ in desired savings arising from the higher interest rate. (If there is an initial fall $\Delta I$ in investment in response to the higher interest rate too, then this becomes $\Delta Y = (\Delta S - \Delta I)/MPS$; and it becomes more complicated still if investment changes in response to income, etc..)

In short, $S=I$ continues to hold because *desired savings depends on income* - which you can't see from the simple, partial equilibrium diagram showing S and I relative to the interest rate. Consumption (and therefore income) will eventually fall enough to cause a decline in savings that restores equilibrium.


---

### Answer by username_3 on 2015-05-07. Score: 0

### Are prices exogenous?

You assumed an exogenous shift in the interest rate. Typically, prices are endogenous in Macroeconomics (and the interest rate is nothing but the price of capital). Absent frictions, prices adjust such that supply equals demand.

That is exactly why your thought experiment doesn't make much sense in the very stylized model: Suppose we somehow raise the interest rate, then supply does not equal demand anymore (unless there is storage, see below). Hence, the [Walrasian auctioneer](http://en.wikipedia.org/wiki/Walrasian_auction) will decrease the interest rate until supply equals demand again - the original point.

### Can there be exogenous shocks to prices?

In a more sophisticated world, exogenous factors such as monetary policy affect the interest rate. Why? Well, because actually savings needn't be investment. For example, if households just hold cash at home and do not spend it, that is savings (defined as income minus consumption), but it is not invested. Similar stories hold for banks that hold cash (or store their money at their central bank).

> **Comments:**
> **username_0**: Consider the following argument: the rise in interest rate caused business to borrow less for investment, and consequently the production of the capital will decrease, which consequently cause the national income to decrease. So people does not have as much income as before when the interest rate has not rose, so their saving is clearly also decreased.        Do you think it is an valid argument?  
> **username_3**: @Kund that depends on your timing assumption. Typically, investment *now* yields additional production *in the future* - there is some delay. Hence, a decrease in investment today will not decrease income today, but rather a decrease in income tomorrow.  
> **username_0**: I mean the investment money now would yields production of capital today. I mean the production of capital instead of what can be produced by the capital in the future.  
> **username_2**: @username_3, the 1st half of your answer is great, but I have to strongly disagree with the 2nd half. In a closed economy, when savings and investment are defined properly (combining public and private sectors), S=I will always hold. Your example of households and cash isn't right: if households in the aggregate demand more cash, then either the central bank will trade them that cash for other assets (in which case private & public savings are both unchanged), or government will issue more liabilities (in which case private savings rise and public savings fall, with no net _aggregate_ change).  
> **username_2**: @username_0, yes, I think your argument has the right idea, and I make a more elaborate version of this argument in my answer. When a rise in interest rates causes less desired investment and more desired savings (meaning less desired consumption), income will fall and keep falling until savings decline enough to restore equilibrium.  
> **username_3**: @username_2 I'll might answer more as I'm in a hurry, but two points (i) I should've mentioned sticky prices. My answer was completely out of NK context or monetary policy: I see the issue with cash holdings, but they should've just been an example for storage devices. Now, for example, if inventories (which are technically storage device) are *defined* to be investment, and if similar storage devices all are defined as such, then *when savings and investment are defined properly, S=I will always hold* might be tautologically true.  
> **username_2**: @username_3, S=I is in one sense a tautology, but it's a tautology in the same sense that income=production or eq supply=eq demand are, and like them it has pretty profound consequences. Note the national accounts are set up to enforce S=I, albeit adjusted for the open economy: net savings equal net investment plus the current account (up to the GDP/GDI statistical discrepancy); see [here](http://research.stlouisfed.org/fred2/graph/?g=1aOU).  

---

