# Example of production function with negative returns with respect one input

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?

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 $$\beta_p<0$$.

• 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. Jul 11, 2021 at 16:20
• I mean, if P is exogenously assigned due to government regulations (like safety inspections), it would be > 0. Jul 11, 2021 at 16:23
• 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). Jul 11, 2021 at 16:25
• 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 Jul 12, 2021 at 9:24