<p>What you have proven is that $$f(tx_1, tx_2) &gt; tg(x_1,x_2)$$</p>

<p>which translates "scaled $f$ is higher than <em>some other</em>, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise.</p>

<p>You can go in reverse, starting from </p>

<p>$$tf(x_1,x_2) = t^{0.5}t^{0.5}f(x_1,x_2)=...$$</p>

<p>and compare to $f(tx_1, tx_2)$. </p>

<p>Also beware about the following: what are the permissible values for $t$ when returns are not constant? Are you sure we should obtain the result for $t&gt;0$, or maybe we examine the property for a smaller interval?</p>

CC BY-SA 4.0
2018-07-05T13:18:57.517Z
For IRS and DRS the constant t should be greater than 1, got it, thank you! However, since I've been asked to find out the returns to scale and thus when I still don't know whether it is CRS or otherwise, in that case, what should be my assumption about the constant t be?
Chd
CC BY-SA 4.0
2018-07-05T13:25:06.267Z
@Clarissa Is it possible for this function to exhibit CRS?
Alecos Papadopoulos
CC BY-SA 4.0
2018-07-05T13:49:25.367Z
Err no, I got your point, thanks for replying @Alecos Paradopoulos
Chd
CC BY-SA 4.0
2018-07-05T02:49:15.507Z
2018-07-05T13:21:47.310Z
2018-07-05T13:21:47.310Z
Alecos Papadopoulos
<p>Find the returns to scale for he following production function- $(x_1 + 1)^.5 (x_2)^.5$</p>

<p>My attempt,
Let $f(x_1,x_2)=(x_1 + 1)^.5 (x_2)^.5$
And $g(x_1,x_2)=(x_1)^.5 (x_2)^.5$
Now,
$(x_1 + 1)^.5 (x_2)^.5$ 
$&gt;$
$(x1)^.5 (x_2)^.5$
Therefore,
$f(x_1,x_2)&gt;g(x_1,x_2)$
Or, $f(tx_1,tx_2)&gt;g(tx_1,tx_2)$ for some t>0
$&gt;(tx_1)^.5 (tx_2)^.5$
$=t(x_1)^.5 (x_2)^.5$
$=tg(x_1,x_2)$
The function g exhibits constant returns to scale. The function f being bigger than it should exhibit increasing returns to scale. The answer given, however, is decreasing returns to scale. Where am I wrong? Thank you in advance.</p>

<microeconomics><economies-of-scale>
finding the returns to scale of a production function
Chd
# finding the returns to scale of a production function

(Asked by: username_0 on 2018-07-04)

Find the returns to scale for he following production function- $(x\_1 + 1)^.5 (x\_2)^.5$

My attempt,
Let $f(x\_1,x\_2)=(x\_1 + 1)^.5 (x\_2)^.5$
And $g(x\_1,x\_2)=(x\_1)^.5 (x\_2)^.5$
Now,
$(x\_1 + 1)^.5 (x\_2)^.5$
$>$
$(x1)^.5 (x\_2)^.5$
Therefore,
$f(x\_1,x\_2)>g(x\_1,x\_2)$
Or, $f(tx\_1,tx\_2)>g(tx\_1,tx\_2)$ for some t>0
$>(tx\_1)^.5 (tx\_2)^.5$
$=t(x\_1)^.5 (x\_2)^.5$
$=tg(x\_1,x\_2)$
The function g exhibits constant returns to scale. The function f being bigger than it should exhibit increasing returns to scale. The answer given, however, is decreasing returns to scale. Where am I wrong? Thank you in advance.

**Tags:** microeconomics, economies-of-scale


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## 1 answers

### Answer by username_1 on 2018-07-05. Score: 3 | Accepted answer

What you have proven is that $$f(tx\_1, tx\_2) > tg(x\_1,x\_2)$$

which translates "scaled $f$ is higher than *some other*, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise.

You can go in reverse, starting from

$$tf(x\_1,x\_2) = t^{0.5}t^{0.5}f(x\_1,x\_2)=...$$

and compare to $f(tx\_1, tx\_2)$.

Also beware about the following: what are the permissible values for $t$ when returns are not constant? Are you sure we should obtain the result for $t>0$, or maybe we examine the property for a smaller interval?

> **Comments:**
> **username_0**: For IRS and DRS the constant t should be greater than 1, got it, thank you! However, since I've been asked to find out the returns to scale and thus when I still don't know whether it is CRS or otherwise, in that case, what should be my assumption about the constant t be?  
> **username_1**: @Clarissa Is it possible for this function to exhibit CRS?  
> **username_0**: Err no, I got your point, thanks for replying @username_1 Paradopoulos  

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