<p>Certainly the economy is not a "zero sum game" (whatever that means). There has historically been a positive aggregate net return on capital investment - there doesn't have to be a loser for every winner.</p>

<p>That said, economists generally believe that as the stock of aggregate savings increases, the return will go down; this is a consequence of diminishing returns to capital. Hence it's not possible for everyone to save an <em>unlimited</em> amount at 4%.</p>

<p>To be a little more concrete, economists often use a constant-returns-to-scale production function $F(K,L)$, taking capital $K$ and labor $L$ as inputs, as a first-pass way of thinking about the world. In this simple model, net aggregate savings equal capital $K$. Holding $L$ constant, the extent to which more $K$ will push down the net return to capital $r=F_K-\delta$ (where $F_K$ is the marginal product of capital and $\delta$ is the depreciation rate) depends on the <a href="http://en.wikipedia.org/wiki/Elasticity_of_substitution" rel="noreferrer"><em>elasticity of substitution</em></a> of the function $F$. </p>

<p>With a high elasticity of substitution, $K$ can increase substantially without $r$ falling very much, as the economy continues to find productive applications for capital despite its relative abundance; with a low elasticity of substitution, a rise in $K$ can result in a large drop in $r$.</p>

<hr>

<p><strong>Example.</strong> A common form for $F$ is <em>Cobb-Douglas</em>, which corresponds to a constant elasticity of substitution of 1. If the gross capital share of production is $0.35$, then we write $F(K,L) = K^{0.35} L^{0.65}$. The marginal product of capital here is $F_K(K,L) = 0.35 \cdot (K/L)^{-0.65}$, implying that the elasticity of $F_K$ with respect to $K$ is -0.65.</p>

<p>Now, if we suppose that initially $r=0.04$ and $\delta=0.06$, then the elasticity of $r=F_K-\delta$ with respect to $K$ is $-0.65\cdot (0.1/0.04)=-1.625$. This means that if we increase capital $K$ by 10%, the net return $r$ will decrease by approximately 16.25%, or $4\times .1625 = .65$ percentage points, from 4% to 3.35%. This is a pretty substantial decrease!</p>

<p>Indeed, since $1.625 &gt; 1$, in this case net capital income $rK$ to capital decreases as $K$ goes up; remarkably enough, if savers accumulate more, they earn less, as the decline in the net return per unit of capital overwhelms the increase in capital.</p>

CC BY-SA 3.0
2014-12-28T17:19:48.247Z
I am not sure I understand all the math behind, BUT why is the δ=0.06? a depreciation rate of 6% that means the asset got 16.667 years to lose all its value. am I right?
dreamoki
CC BY-SA 3.0
2014-12-28T18:38:06.730Z
Yes, although I'm assuming geometric depreciation rather than straight-line depreciation, so it's more accurate to say that the average lifetime of capital is 16.667 years rather than that it will lose all its value in 16.667 years. I picked $\delta=0.06$ somewhat randomly, but 0.064 is actually is the average depreciation rate for fixed assets and durable goods in the US: see [line 1 here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=86) divided by [line 1 here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=16).
nominally rigid
CC BY-SA 3.0
2014-12-28T18:56:20.840Z
is this rule true for every economy? or just the regular economy - for example, if we had an economy of 10 people who saved 10 USD every month (100 USD total) and raise their capital saving in 10% next month it will too influence the return rate in this substantial manner? what affects this outcome really?
dreamoki
CC BY-SA 3.0
2014-12-28T19:10:08.437Z
Well, certainly the exact form of $F$ and its current elasticity of substitution differs depending on the economy and society. But I think the qualitative idea of diminishing returns is pretty universal, and that you might often get substantial magnitudes like this. One nearly-universal fact is that the value of the capital stock is mostly in structures (see [here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=16) for US) - and it's pretty natural that your second house or office or power plant is less important than your first.
nominally rigid
CC BY-SA 3.0
2014-12-27T07:05:40.493Z
2014-12-27T07:05:40.493Z
nominally rigid
<p>Yes, it's possible, as long as the money is invested in ways that increase production by 4%. The economy is certainly not a zero sum game -- if it was, then it couldn't grow.</p>

CC BY-SA 3.0
2014-12-24T20:15:53.547Z
Mike please comment on the comments of my original post.
dreamoki
CC BY-SA 3.0
2014-12-26T15:45:33.867Z
@dreamoki - this question has been moved to this site, the original posting is closed to new answers or comments.
JTP - Apologise to Monica
CC BY-SA 3.0
2014-12-24T12:16:16.913Z
2014-12-24T12:16:16.913Z
Mike Scott
<p>Theoretically speaking, if all the Earth's inhabitants were to save money and invest, is it possible for everybody to get, let's say, a 4% yearly return for everyone?</p>

<p>Is the economy a zero sum game (where if someone gains, others lose) or not?</p>

CC BY-SA 3.0
2014-12-24T12:17:41.790Z
If everybody saves and invest who will do the spending ? The saving and spending has to go in tandem.
DumbCoder
CC BY-SA 3.0
2014-12-24T14:05:39.003Z
Read it carefully. Investment means spending further down the line. You only factor savings and spending ? What about other economic factors like inflation ?
DumbCoder
CC BY-SA 3.0
2014-12-28T04:39:36.030Z
You're asking whether it's possible for everyone to plant a tree?
Steven Landsburg
CC BY-SA 3.0
2014-12-28T16:37:52.703Z
@StevenL - I guess technically it is possible for everyone to plant a tree :) but it contradicts what nominally rigid is saying below...
dreamoki
CC BY-SA 3.0
2014-12-28T17:08:23.850Z
@dreamoki:  Nominally Rigid is saying that we can't all plant trees in the most fertile ground, so if more of us plant trees, then some of us are going to have to plant them where they don't thrive as well.  That doesn't mean we can't all get positive returns.
Steven Landsburg
CC BY-SA 3.0
2014-12-28T17:37:02.140Z
Yes but the example he gave showed that if you added more 10% capital the net return would significantly diminish. and I am a looking for an example that it can be stable even with large amounts of people.
dreamoki
<macroeconomics>
Is the economy a zero-sum game?
dreamoki
# Is the economy a zero-sum game?

(Asked by: username_0 on 2014-12-24)

Theoretically speaking, if all the Earth's inhabitants were to save money and invest, is it possible for everybody to get, let's say, a 4% yearly return for everyone?

Is the economy a zero sum game (where if someone gains, others lose) or not?

**Tags:** macroeconomics
> **Comments:**
> **username_1**: If everybody saves and invest who will do the spending ? The saving and spending has to go in tandem.  
> **username_1**: Read it carefully. Investment means spending further down the line. You only factor savings and spending ? What about other economic factors like inflation ?  
> **username_2**: You're asking whether it's possible for everyone to plant a tree?  
> **username_0**: @StevenL - I guess technically it is possible for everyone to plant a tree :) but it contradicts what nominally rigid is saying below...  
> **username_2**: @dreamoki:  Nominally Rigid is saying that we can't all plant trees in the most fertile ground, so if more of us plant trees, then some of us are going to have to plant them where they don't thrive as well.  That doesn't mean we can't all get positive returns.  
> **username_0**: Yes but the example he gave showed that if you added more 10% capital the net return would significantly diminish. and I am a looking for an example that it can be stable even with large amounts of people.  

---

## 2 answers

### Answer by username_4 on 2014-12-24. Score: 6 | Accepted answer

Yes, it's possible, as long as the money is invested in ways that increase production by 4%. The economy is certainly not a zero sum game -- if it was, then it couldn't grow.

> **Comments:**
> **username_0**: Mike please comment on the comments of my original post.  
> **username_5**: @dreamoki - this question has been moved to this site, the original posting is closed to new answers or comments.  

---

### Answer by username_3 on 2014-12-27. Score: 9

Certainly the economy is not a "zero sum game" (whatever that means). There has historically been a positive aggregate net return on capital investment - there doesn't have to be a loser for every winner.

That said, economists generally believe that as the stock of aggregate savings increases, the return will go down; this is a consequence of diminishing returns to capital. Hence it's not possible for everyone to save an *unlimited* amount at 4%.

To be a little more concrete, economists often use a constant-returns-to-scale production function $F(K,L)$, taking capital $K$ and labor $L$ as inputs, as a first-pass way of thinking about the world. In this simple model, net aggregate savings equal capital $K$. Holding $L$ constant, the extent to which more $K$ will push down the net return to capital $r=F\_K-\delta$ (where $F\_K$ is the marginal product of capital and $\delta$ is the depreciation rate) depends on the [*elasticity of substitution*](http://en.wikipedia.org/wiki/Elasticity_of_substitution) of the function $F$.

With a high elasticity of substitution, $K$ can increase substantially without $r$ falling very much, as the economy continues to find productive applications for capital despite its relative abundance; with a low elasticity of substitution, a rise in $K$ can result in a large drop in $r$.

---

**Example.** A common form for $F$ is *Cobb-Douglas*, which corresponds to a constant elasticity of substitution of 1. If the gross capital share of production is $0.35$, then we write $F(K,L) = K^{0.35} L^{0.65}$. The marginal product of capital here is $F\_K(K,L) = 0.35 \cdot (K/L)^{-0.65}$, implying that the elasticity of $F\_K$ with respect to $K$ is -0.65.

Now, if we suppose that initially $r=0.04$ and $\delta=0.06$, then the elasticity of $r=F\_K-\delta$ with respect to $K$ is $-0.65\cdot (0.1/0.04)=-1.625$. This means that if we increase capital $K$ by 10%, the net return $r$ will decrease by approximately 16.25%, or $4\times .1625 = .65$ percentage points, from 4% to 3.35%. This is a pretty substantial decrease!

Indeed, since $1.625 > 1$, in this case net capital income $rK$ to capital decreases as $K$ goes up; remarkably enough, if savers accumulate more, they earn less, as the decline in the net return per unit of capital overwhelms the increase in capital.

> **Comments:**
> **username_0**: I am not sure I understand all the math behind, BUT why is the δ=0.06? a depreciation rate of 6% that means the asset got 16.667 years to lose all its value. am I right?  
> **username_3**: Yes, although I'm assuming geometric depreciation rather than straight-line depreciation, so it's more accurate to say that the average lifetime of capital is 16.667 years rather than that it will lose all its value in 16.667 years. I picked $\delta=0.06$ somewhat randomly, but 0.064 is actually is the average depreciation rate for fixed assets and durable goods in the US: see [line 1 here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=86) divided by [line 1 here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=16).  
> **username_0**: is this rule true for every economy? or just the regular economy - for example, if we had an economy of 10 people who saved 10 USD every month (100 USD total) and raise their capital saving in 10% next month it will too influence the return rate in this substantial manner? what affects this outcome really?  
> **username_3**: Well, certainly the exact form of $F$ and its current elasticity of substitution differs depending on the economy and society. But I think the qualitative idea of diminishing returns is pretty universal, and that you might often get substantial magnitudes like this. One nearly-universal fact is that the value of the capital stock is mostly in structures (see [here](http://bea.gov/iTable/iTableHtml.cfm?reqid=10&step=3&isuri=1&1003=16) for US) - and it's pretty natural that your second house or office or power plant is less important than your first.  

---

