<p>Production functions are defined without specific values for parameters, so they all could if you impose that the logical parameter implies a negative return.</p>
<p>For example, consider a Cobb-Douglas production function of capital and labor,</p>
<p><span class="math-container">$Y=\beta_0 K^{\beta_k}L^{\beta_l}\omega \varepsilon$</span></p>
<p>where <span class="math-container">$\omega$</span> denotes firm-observed productivity and <span class="math-container">$\varepsilon$</span> is an idiosyncratic shock. If you wanted to augment this with an input that descreases prodcution, maybe <span class="math-container">$P$</span>, you'd just write,</p>
<p><span class="math-container">$Y=\beta_0 K^{\beta_k}L^{\beta_l}P^{\beta_p}\omega \varepsilon$</span></p>
<p>and impose that <span class="math-container">$\beta_p&lt;0$</span>.</p>

CC BY-SA 4.0
2021-07-11T16:20:10.207Z
Thank you, that was easy. But then we might as well remove this factor from the production function because a rational firm would never choose $P>0$? I was thinking more of a setup where under certain combinations of factors or at certain levels of production negative total returns kick on.
Papayapap
CC BY-SA 4.0
2021-07-11T16:23:39.747Z
I mean, if P is exogenously assigned due to government regulations (like safety inspections), it would be > 0.
Michael Gmeiner
CC BY-SA 4.0
2021-07-11T16:25:12.497Z
Most production functions exhibit decreasing returns to individual inputs, but are not in functional forms where it would be negative, just that returns approach 0. (e.g. Cobb-Douglas).
Michael Gmeiner
CC BY-SA 4.0
2021-07-12T09:24:44.567Z
Another reason why P might be positive is if it's a joint effect of the use of one of the factors of production. E.g. when the use of capital increases pollution (P) which has an effect on output
Maarten Punt
CC BY-SA 4.0
2021-07-11T16:12:41.557Z
2021-07-11T16:12:41.557Z
Michael Gmeiner
<p>Are there examples of production functions where increasing the input of one factor and keeping the other factor constant leads to reductions in total production?</p>

<microeconomics><production-function>
Example of production function with negative returns with respect one input
Papayapap
# Example of production function with negative returns with respect one input

(Asked by: username_0 on 2021-07-11)

Are there examples of production functions where increasing the input of one factor and keeping the other factor constant leads to reductions in total production?

**Tags:** microeconomics, production-function

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## 1 answers

### Answer by username_1 on 2021-07-11. Score: 3 | Accepted answer

Production functions are defined without specific values for parameters, so they all could if you impose that the logical parameter implies a negative return.

For example, consider a Cobb-Douglas production function of capital and labor,

$Y=\beta\_0 K^{\beta\_k}L^{\beta\_l}\omega \varepsilon$

where $\omega$ denotes firm-observed productivity and $\varepsilon$ is an idiosyncratic shock. If you wanted to augment this with an input that descreases prodcution, maybe $P$, you'd just write,

$Y=\beta\_0 K^{\beta\_k}L^{\beta\_l}P^{\beta\_p}\omega \varepsilon$

and impose that $\beta\_p<0$.

> **Comments:**
> **username_0**: Thank you, that was easy. But then we might as well remove this factor from the production function because a rational firm would never choose $P>0$? I was thinking more of a setup where under certain combinations of factors or at certain levels of production negative total returns kick on.  
> **username_1**: I mean, if P is exogenously assigned due to government regulations (like safety inspections), it would be > 0.  
> **username_1**: Most production functions exhibit decreasing returns to individual inputs, but are not in functional forms where it would be negative, just that returns approach 0. (e.g. Cobb-Douglas).  
> **username_2**: Another reason why P might be positive is if it's a joint effect of the use of one of the factors of production. E.g. when the use of capital increases pollution (P) which has an effect on output  

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