# Total Factor Productivity and Leontief Production Function

Just a bit of a conceptual question regarding the Leontief production function and the concept of total factor productivity (TFP).

In particular, I wondered if these two concepts are at odds?

• The Leontief production function assumes output is proportional to input. Output is thus explained by inputs.

• Yet, TFP is the portion of output unexplained by inputs. Therefore, there is some hiddenness regarding TFP beyond simple proportionality.

As such, I wondered if there is any room for the concept of TFP in a Leontief production (and thus input-output analysis), or if TFP is only suitable with more flexible Cobb-Douglas-like production functions?

Would be good to hear your thoughts.

• How exactly do "Cobb-Douglas-like production functions" explain something "unexplained by inputs"? Feb 13 at 6:12
• Hi @ Michael. I feel that's a mischaracterisation of what I said above. I never said Cobb-Douglas "explain something unexplained by inputs." Instead, I was asking if more flexible functional forms are a pre-requisite when it comes to accounting for TFP. However, I've since found some literature on Leontief and TFP. Miller and Blair: Input-Output, Chap 15. Feb 13 at 7:53
• Proportionality comes from the fact that all production plans will be proportional with a Leontief production function. Your question asked nothing like that, you changed it after the initial question was answered. Feb 13 at 14:12
• Hi @MichaelGreinecker. The edit history should be transparent. I never changed it in a way that would affect interpretation of my question. Instead, I emphasised the point on the hiddenness of the TFP concept. The first bullet point, and the questions remain the same in both versions of the question. However, I am very happy to take withdraw this question if there is a means too. Feb 13 at 15:35
• I do think it would be better to state that a part was edited in the question. On the substantive point, there really is not much of a difference to how TFP works with CD-production functions: $(aL)^\alpha(bK)^\beta=a^\alpha b^\beta L^\alpha K^\beta$. Feb 13 at 16:16

Every production function "explains output by inputs." But you can still model changes in TFP using Leontief production functions. Let, for example, $$f(K,L)=\min\{\kappa K,\lambda L\}$$ with $$\kappa$$ and $$\lambda$$ strictly positive. You can write this as $$f(K,L)=\min\{ A \kappa K, A\lambda L\}=A \min\{\kappa K,\lambda L\}$$ with $$A=1$$. You can interpret $$A$$ as total factor productivity and consider changes in $$A$$. The point is that proportional changes in both $$\kappa$$ and $$\lambda$$ correspond to changes in total factor productivity.