# Is the economy a zero-sum game?

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?

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If everybody saves and invest who will do the spending ? The saving and spending has to go in tandem. –  DumbCoder Dec 24 '14 at 12:17
see what Mike Scott wrote below. it is possible, not all your spending are cut to savings... like everybody do. –  dreamoki Dec 24 '14 at 12:49
Read it carefully. Investment means spending further down the line. You only factor savings and spending ? What about other economic factors like inflation ? –  DumbCoder Dec 24 '14 at 14:05
You're asking whether it's possible for everyone to plant a tree? –  Steven Landsburg Dec 28 '14 at 4:39
@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 Dec 28 '14 at 17:08

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.

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Mike please comment on the comments of my original post. –  dreamoki Dec 24 '14 at 20:15
@dreamoki - this question has been moved to this site, the original posting is closed to new answers or comments. –  JoeTaxpayer Dec 26 '14 at 15:45

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 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.

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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 Dec 28 '14 at 17:19
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 divided by line 1 here. –  nominally rigid Dec 28 '14 at 18:38
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 Dec 28 '14 at 18:56
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 for US) - and it's pretty natural that your second house or office or power plant is less important than your first. –  nominally rigid Dec 28 '14 at 19:10