<p>Per our comments above, I believe the expression you have is incorrect.  Your confusion is worsened by some poor terminology use.</p>

<p>Gross return can be written as <span class="math-container">$R = 1 + \frac{a}{w}$</span> where <span class="math-container">$w$</span> is initial wealth (or more accurately, the initial investment) and <span class="math-container">$a$</span> is new cash flow generated by the investment.  So for example, with <span class="math-container">$w = 100$</span> and <span class="math-container">$a = 5$</span>, the gross return is <span class="math-container">$R = 1 + \frac{5}{100}$</span> = 1.05, representing a 5% return on $100.</p>

<p>What you call the "gross return of wealth" is, I think, just the derivative of <span class="math-container">$R$</span> with respect to <span class="math-container">$w$</span>: <span class="math-container">$\frac{dR}{dw} = -\frac{a}{w^2}$</span>.  The question is, why is this negative?  Because what this quantity tells you is how your gross return changes when your initial investment grows, holding the new cash flow constant.</p>

<p>It really just follows from the arithmetic of proportions. <em>Return</em> is a proportion while <em>wealth</em> is a number.  As wealth <span class="math-container">$w$</span> increases, the return from a new cash flow <span class="math-container">$a$</span> is reduced, because it is a smaller proportion of a larger whole.  So in the example above, if we double our investment to <span class="math-container">$w = 200$</span> but we keep <span class="math-container">$a = 5$</span>, then the return is <span class="math-container">$R = 1 + \frac{5}{200} = 1.025$</span>.  Because we had to invest more to get the same amount of new dollars, our <em>return</em> is smaller.</p>

CC BY-SA 4.0
2019-11-20T21:11:26.713Z
2019-11-20T21:11:26.713Z
heh
<p>I have a short question: </p>

<p>Why  does the gross return of wealth decrease as wealth increases?</p>

<p>Why do the gross return of wealth and wealth move in opposite direction?</p>

<p><span class="math-container">$$R=1+(a/w^2)$$</span> where a is non-negative parameter.</p>

CC BY-SA 4.0
2019-11-20T18:41:36.443Z
Are you sure your expression is correct?  I believe you should have a minus sign in there, not a plus.
heh
CC BY-SA 4.0
2019-11-20T18:56:47.923Z
In the question, this equation is given like that. So I don't know. If what you said is true, please can you refer it by a book? @heh
studentp
CC BY-SA 4.0
2019-11-20T19:19:52.230Z
Do you know calculus?
heh
CC BY-SA 4.0
2019-11-20T20:52:15.657Z
@heh yes I know of course
studentp
CC BY-SA 4.0
2019-11-20T20:55:08.130Z
OK: I'll write out an answer below that should help.
heh
<macroeconomics><self-study><wealth>
Why do the gross return of wealth and wealth move in opposite direction?
studentp
# Why do the gross return of wealth and wealth move in opposite direction?

(Asked by: username_0 on 2019-11-20)

I have a short question:

Why does the gross return of wealth decrease as wealth increases?

Why do the gross return of wealth and wealth move in opposite direction?

$$R=1+(a/w^2)$$ where a is non-negative parameter.

**Tags:** macroeconomics, self-study, wealth

> **Comments:**
> **username_1**: Are you sure your expression is correct?  I believe you should have a minus sign in there, not a plus.  
> **username_0**: In the question, this equation is given like that. So I don't know. If what you said is true, please can you refer it by a book? @username_1  
> **username_1**: Do you know calculus?  
> **username_0**: @username_1 yes I know of course  
> **username_1**: OK: I'll write out an answer below that should help.  

---

## 1 answers

### Answer by username_1 on 2019-11-20. Score: 2 | Accepted answer

Per our comments above, I believe the expression you have is incorrect. Your confusion is worsened by some poor terminology use.

Gross return can be written as $R = 1 + \frac{a}{w}$ where $w$ is initial wealth (or more accurately, the initial investment) and $a$ is new cash flow generated by the investment. So for example, with $w = 100$ and $a = 5$, the gross return is $R = 1 + \frac{5}{100}$ = 1.05, representing a 5% return on $100.

What you call the "gross return of wealth" is, I think, just the derivative of $R$ with respect to $w$: $\frac{dR}{dw} = -\frac{a}{w^2}$. The question is, why is this negative? Because what this quantity tells you is how your gross return changes when your initial investment grows, holding the new cash flow constant.

It really just follows from the arithmetic of proportions. *Return* is a proportion while *wealth* is a number. As wealth $w$ increases, the return from a new cash flow $a$ is reduced, because it is a smaller proportion of a larger whole. So in the example above, if we double our investment to $w = 200$ but we keep $a = 5$, then the return is $R = 1 + \frac{5}{200} = 1.025$. Because we had to invest more to get the same amount of new dollars, our *return* is smaller.


---

