<p>I have a production function of the form <span class="math-container">$f(x_1,x_2) = x_1^a x_2^b$</span> and I am trying to figure out what the global returns to scale would be given that <span class="math-container">$a,b \in (0,1)$</span>.</p>
<p>This production function is homogenous of degree <span class="math-container">$a+b$</span> so if both <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are decimals then we could have:</p>
<ul>
<li><span class="math-container">$(a,b,a+b) = (0.6, 0.7, 1.3)$</span> (increasing returns)</li>
<li><span class="math-container">$(a,b,a+b) = (0.2, 0.5, 0.7)$</span> (decreasing returns)</li>
<li><span class="math-container">$(a,b,a+b) = (0.45, 0.55, 1)$</span> (constant returns)</li>
</ul>
<p>This would make it difficult to determine exactly what the returns to scale are. Can anyone help with this?</p>

CC BY-SA 4.0
2023-05-30T03:06:35.427Z
Does [this](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions) help you?
solow supremacy
CC BY-SA 4.0
2023-05-31T06:12:27.087Z
Using $f(\lambda x_1, \lambda x_2) = \lambda^{a+b} f(x_1, x_2) \gtreqqless \lambda f(x_1, x_2)$ $(\forall \ \lambda > 1)$ and the "formal definition" written on the [Wiki page](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions), are you able to solve the question? Hint: It depends on the value of $a+b$.
solow supremacy
<microeconomics><production-function><producer-theory>
Global returns to scale
Debbie
# Global returns to scale

(Asked by: username_0 on 2023-05-29)

I have a production function of the form $f(x\_1,x\_2) = x\_1^a x\_2^b$ and I am trying to figure out what the global returns to scale would be given that $a,b \in (0,1)$.

This production function is homogenous of degree $a+b$ so if both $a$ and $b$ are decimals then we could have:

* $(a,b,a+b) = (0.6, 0.7, 1.3)$ (increasing returns)
* $(a,b,a+b) = (0.2, 0.5, 0.7)$ (decreasing returns)
* $(a,b,a+b) = (0.45, 0.55, 1)$ (constant returns)

This would make it difficult to determine exactly what the returns to scale are. Can anyone help with this?

**Tags:** microeconomics, production-function, producer-theory

> **Comments:**
> **username_1**: Does [this](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions) help you?  
> **username_1**: Using $f(\lambda x_1, \lambda x_2) = \lambda^{a+b} f(x_1, x_2) \gtreqqless \lambda f(x_1, x_2)$ $(\forall \ \lambda > 1)$ and the "formal definition" written on the [Wiki page](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions), are you able to solve the question? Hint: It depends on the value of $a+b$.  

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## 0 answers

