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

CC BY-SA 4.0
2023-02-12T15:23:55.800Z
Thanks @Michael Greinecker. I appreciate your feedback. Have you seen this applied in I-O style analysis before by chance? I couldn't find any sources. So I started to doubt whether the two concepts were compatible.
EB3112
CC BY-SA 4.0
2023-02-12T15:26:38.093Z
@EB3112 That is not an area I'm really familiar with. So while I haven't seen this, it doesn't mean much.
Michael Greinecker
CC BY-SA 4.0
2023-02-12T15:10:53.987Z
2023-02-12T15:10:53.987Z
Michael Greinecker
<p>Just a bit of a conceptual question regarding the Leontief production function and the concept of total factor productivity (TFP).</p>
<p>In particular, I wondered if these two concepts are at odds?</p>
<ul>
<li><p>The Leontief production function assumes output is proportional to input. Output is thus explained by inputs.</p>
</li>
<li><p>Yet, TFP is the portion of output <em>unexplained</em> by inputs. Therefore, there is some hiddenness regarding TFP beyond simple proportionality.</p>
</li>
</ul>
<p>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?</p>
<p>Would be good to hear your thoughts.</p>

CC BY-SA 4.0
2023-02-13T06:12:00.697Z
How exactly do "Cobb-Douglas-like production functions" explain something "unexplained by inputs"?
Michael Greinecker
CC BY-SA 4.0
2023-02-13T07:53:09.353Z
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.
EB3112
CC BY-SA 4.0
2023-02-13T14:12:56.267Z
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.
Michael Greinecker
CC BY-SA 4.0
2023-02-13T15:35:17.260Z
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.
EB3112
CC BY-SA 4.0
2023-02-13T16:16:15.027Z
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: $(aL)^\alpha(bK)^\beta=a^\alpha b^\beta L^\alpha K^\beta$.
Michael Greinecker
<production-function><productivity>
Total Factor Productivity and Leontief Production Function
EB3112
# Total Factor Productivity and Leontief Production Function

(Asked by: username_0 on 2023-02-12)

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.

**Tags:** production-function, productivity

> **Comments:**
> **username_1**: How exactly do "Cobb-Douglas-like production functions" explain something "unexplained by inputs"?  
> **username_0**: 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.  
> **username_1**: 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.  
> **username_0**: 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.  
> **username_1**: 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: $(aL)^\alpha(bK)^\beta=a^\alpha b^\beta L^\alpha K^\beta$.  

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## 1 answers

### Answer by username_1 on 2023-02-12. Score: 1

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.

> **Comments:**
> **username_0**: Thanks @username_1. I appreciate your feedback. Have you seen this applied in I-O style analysis before by chance? I couldn't find any sources. So I started to doubt whether the two concepts were compatible.  
> **username_1**: @username_0 That is not an area I'm really familiar with. So while I haven't seen this, it doesn't mean much.  

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