<p>I figured it out:</p>
<p>The first-order condition of the cost minimization problem for, say, material inputs <span class="math-container">$m_{it}$</span> gives:</p>
<p><span class="math-container">$ \lambda \frac{\partial F}{\partial M} = P_M $</span></p>
<p>Where F is the production function, <span class="math-container">$P_M$</span> the material input prices. Multiply by <span class="math-container">$\frac{M}{F}$</span> and rearrange,</p>
<p><span class="math-container">$ \lambda = \frac{P_M M}{\beta_M F}  $</span>,</p>
<p>where <span class="math-container">$\beta_M$</span> is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.</p>

CC BY-SA 4.0
2021-03-29T12:52:09.787Z
It should be $\lambda = \frac{P_M M}{\beta_M F}$, and in the end it returns a formula for marginal cost that is not very useful. Specifically, it does not return marginal cost as a function of the sum of elasticities and therefore does not establish a connection to the returns to scale. There is no such connection, since by scaling up all inputs you typically violate cost minimization.
VARulle
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2021-03-29T21:10:18.990Z
You're right, thanks. Typo in the formula, and one can only get the short-run returns to scale. Corrected.
hrrrrrr5602
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2021-03-29T21:11:00.353Z
I don't agree with the "not very useful", since I was mostly looking for a relation between MC and output elasticities.
hrrrrrr5602
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2021-03-30T11:49:44.083Z
Hmm, but it was formulated as if you were looking for a relation between MC and the *sum* of output elasticities. (That's why I asked about that on March 25.) I'm also skeptical about the "short-run returns to scale" - what's that? I still believe the formula is not very useful, since the elasticities alone don't help, as you also need to know the expansion path $M_i(F)$ to calculate MC for given output and prices.
VARulle
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2021-04-01T21:31:28.683Z
The short-run returns to scale or short-run elasticity of scale tell you how output changes in response to a simultaneous change in both variable inputs. See, e.g. Chambers (1988, pp 69-74). I do not see the relevance of your expansion path comment. The marginal cost can be calculated from the cost function as depending only on input prices and production function inputs.
hrrrrrr5602
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2021-04-02T11:49:06.613Z
That's my point: You need the production function inputs. Knowing output, prices and elasticities is not sufficient. Knowing output *and* the corresponding optimal inputs just means knowing the expansion path.
VARulle
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2021-04-02T20:14:58.670Z
The fact that you need to know the production inputs is implicit in having F in the formula, since F is a function F(X_1, X_2, ..., X_n) where X_i, i=1,..,n are the production function inputs. Also, I did not ask for a relation between marginal cost and output elasticities alone.
hrrrrrr5602
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2021-04-03T22:22:50.303Z
It's not implicit, since F is not injective. Knowing the values of the X_i gives you the value of F,  but not the other way round.
VARulle
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2021-04-03T22:37:22.493Z
Anyway, I find both the question and the answer confusing. The question asks for the relation between MC and the returns to scale (i.e., sum of elasticities). In the comments you say you are just asking for a relation between MC and the elasticities. And the answer (last sentence) says that there is a relation between the sum of "the results" and the returns to scale...
VARulle
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2021-04-06T20:50:32.530Z
I edited the question to read "output elasticities" instead of "sum of output elasticities" to make it less confusing
hrrrrrr5602
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2021-04-06T20:52:42.950Z
F does not need to be injective. We just need to be able to replace F with X_i, which we can do, since as you say, knowing the values of X_i gives you the value of F.
hrrrrrr5602
CC BY-SA 4.0
2021-04-06T20:59:53.047Z
I suggest closing the issue now, unless there are any additional substantive issues to be addressed that are not related to confusion around the question formulation.
hrrrrrr5602
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2021-04-07T07:31:16.613Z
Yeah, let's close it before the mods step in.
VARulle
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2021-03-26T21:46:48.933Z
2021-03-29T21:11:24.457Z
2021-03-29T21:11:24.457Z
hrrrrrr5602
<p>Does anyone know of any results that show the link between marginal cost and the output elasticities analytically? I am looking at production and cost theory books but can't find any results that posit a direct dependence.</p>

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2021-03-25T13:41:03.850Z
Why do you think that marginal costs are a function of the $sum$ of output elasticities?
VARulle
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2021-03-26T21:43:14.290Z
I'm saying the sum because the sum of output elasticities = the returns to scale, which are related to marginal cost, as I say in the first sentence.
hrrrrrr5602
CC BY-SA 4.0
2021-04-06T21:04:10.857Z
Edited the question to remove the part relating to the returns to scale, as it was confusing.
hrrrrrr5602
<elasticity><production-function><cost><marginal>
Relation between marginal cost and output elasticities
hrrrrrr5602
# Relation between marginal cost and output elasticities

(Asked by: username_0 on 2021-03-25)

Does anyone know of any results that show the link between marginal cost and the output elasticities analytically? I am looking at production and cost theory books but can't find any results that posit a direct dependence.

**Tags:** elasticity, production-function, cost, marginal
> **Comments:**
> **username_1**: Why do you think that marginal costs are a function of the $sum$ of output elasticities?  
> **username_0**: I'm saying the sum because the sum of output elasticities = the returns to scale, which are related to marginal cost, as I say in the first sentence.  
> **username_0**: Edited the question to remove the part relating to the returns to scale, as it was confusing.  

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## 1 answers

### Answer by username_0 on 2021-03-26. Score: 3

I figured it out:

The first-order condition of the cost minimization problem for, say, material inputs $m\_{it}$ gives:

$ \lambda \frac{\partial F}{\partial M} = P\_M $

Where F is the production function, $P\_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange,

$ \lambda = \frac{P\_M M}{\beta\_M F} $,

where $\beta\_M$ is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.

> **Comments:**
> **username_1**: It should be $\lambda = \frac{P_M M}{\beta_M F}$, and in the end it returns a formula for marginal cost that is not very useful. Specifically, it does not return marginal cost as a function of the sum of elasticities and therefore does not establish a connection to the returns to scale. There is no such connection, since by scaling up all inputs you typically violate cost minimization.  
> **username_0**: You're right, thanks. Typo in the formula, and one can only get the short-run returns to scale. Corrected.  
> **username_0**: I don't agree with the "not very useful", since I was mostly looking for a relation between MC and output elasticities.  
> **username_1**: Hmm, but it was formulated as if you were looking for a relation between MC and the *sum* of output elasticities. (That's why I asked about that on March 25.) I'm also skeptical about the "short-run returns to scale" - what's that? I still believe the formula is not very useful, since the elasticities alone don't help, as you also need to know the expansion path $M_i(F)$ to calculate MC for given output and prices.  
> **username_0**: The short-run returns to scale or short-run elasticity of scale tell you how output changes in response to a simultaneous change in both variable inputs. See, e.g. Chambers (1988, pp 69-74). I do not see the relevance of your expansion path comment. The marginal cost can be calculated from the cost function as depending only on input prices and production function inputs.  
> **username_1**: That's my point: You need the production function inputs. Knowing output, prices and elasticities is not sufficient. Knowing output *and* the corresponding optimal inputs just means knowing the expansion path.  
> **username_0**: The fact that you need to know the production inputs is implicit in having F in the formula, since F is a function F(X_1, X_2, ..., X_n) where X_i, i=1,..,n are the production function inputs. Also, I did not ask for a relation between marginal cost and output elasticities alone.  
> **username_1**: It's not implicit, since F is not injective. Knowing the values of the X_i gives you the value of F,  but not the other way round.  
> **username_1**: Anyway, I find both the question and the answer confusing. The question asks for the relation between MC and the returns to scale (i.e., sum of elasticities). In the comments you say you are just asking for a relation between MC and the elasticities. And the answer (last sentence) says that there is a relation between the sum of "the results" and the returns to scale...  
> **username_0**: I edited the question to read "output elasticities" instead of "sum of output elasticities" to make it less confusing  
> **username_0**: F does not need to be injective. We just need to be able to replace F with X_i, which we can do, since as you say, knowing the values of X_i gives you the value of F.  
> **username_0**: I suggest closing the issue now, unless there are any additional substantive issues to be addressed that are not related to confusion around the question formulation.  
> **username_1**: Yeah, let's close it before the mods step in.  

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