<p>Seems like the only function <span class="math-container">$f$</span> that fits your description
<span class="math-container">$$
\forall i: \frac{\partial f(\mathbf{x})}{\partial x_i} = c_i
$$</span>
is
<span class="math-container">$$
f(\mathbf{x}) = A + \sum x_i c_i.
$$</span>
(Frequently <span class="math-container">$f(\mathbf{0}) = 0$</span> is assumed. The assumption is referred to as &quot;no free lunch&quot;.)</p>
<p>Then you can apply the definition of returns to scale.</p>

CC BY-SA 4.0
2020-09-23T05:49:24.333Z
2020-09-23T05:49:24.333Z
Giskard
<p>Are there any production function <span class="math-container">$f(x_1,\ldots,x_n)$</span> that is having decreasing returns to scale, given that the marginal product in every input <span class="math-container">$i$</span> in the function <span class="math-container">$f$</span> is constant?</p>

<microeconomics><mathematical-economics><production-function><marginal>
Returns to Scale Microeconomics
Matt
# Returns to Scale Microeconomics

(Asked by: username_0 on 2020-09-23)

Are there any production function $f(x\_1,\ldots,x\_n)$ that is having decreasing returns to scale, given that the marginal product in every input $i$ in the function $f$ is constant?

**Tags:** microeconomics, mathematical-economics, production-function, marginal


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## 1 answers

### Answer by username_1 on 2020-09-23. Score: 3

Seems like the only function $f$ that fits your description
$$
\forall i: \frac{\partial f(\mathbf{x})}{\partial x\_i} = c\_i
$$
is
$$
f(\mathbf{x}) = A + \sum x\_i c\_i.
$$
(Frequently $f(\mathbf{0}) = 0$ is assumed. The assumption is referred to as "no free lunch".)

Then you can apply the definition of returns to scale.


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