<p>The marginal rate of transformation seems to be the one. Or maybe I should say that MRTS is a special case of MRT. When it is assumed that each firm produces one good, <span class="math-container">$F_j(y)=y_{l_j}-f_j(-y_{-l_j})$</span> for some production function <span class="math-container">$f_j$</span>. Then MRTS of a factor <span class="math-container">$l$</span> for factor <span class="math-container">$l'$</span> is <span class="math-container">$\frac{\partial f_j/\partial y_{lj}}{\partial f_j/\partial y_{l'j}}=\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}$</span>, which is MRT of <span class="math-container">$l$</span> for <span class="math-container">$l'$</span>.</p>

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2023-10-17T08:17:12.213Z
2023-10-17T08:17:12.213Z
kenji
<p>I am interested in the definition of the marginal rate of technical substitution (MRTS) under a general setting where the production set of firm <span class="math-container">$j$</span> is <span class="math-container">$Y_j=\{y: F_j(y) \le 0\}$</span>. This formulation allows multiple products to be produced, which makes the usual <span class="math-container">$\frac{MP_K}{MP_L}$</span>-type definition useless. Is there any meaningful extension of the concept of MRTS that deals with multiproduct firms?</p>
<p>cf. Motivation behind this question can be found <a href="https://economics.stackexchange.com/questions/56694/are-the-marginal-conditions-for-pareto-optimality-sufficient">here</a></p>

<microeconomics>
Marginal rate of technical substitution in a general setting
kenji
# Marginal rate of technical substitution in a general setting

(Asked by: username_0 on 2023-10-16)

I am interested in the definition of the marginal rate of technical substitution (MRTS) under a general setting where the production set of firm $j$ is $Y\_j=\{y: F\_j(y) \le 0\}$. This formulation allows multiple products to be produced, which makes the usual $\frac{MP\_K}{MP\_L}$-type definition useless. Is there any meaningful extension of the concept of MRTS that deals with multiproduct firms?

cf. Motivation behind this question can be found [here](https://economics.stackexchange.com/questions/56694/are-the-marginal-conditions-for-pareto-optimality-sufficient)

**Tags:** microeconomics

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## 1 answers

### Answer by username_0 on 2023-10-17. Score: 1 | Accepted answer

The marginal rate of transformation seems to be the one. Or maybe I should say that MRTS is a special case of MRT. When it is assumed that each firm produces one good, $F\_j(y)=y\_{l\_j}-f\_j(-y\_{-l\_j})$ for some production function $f\_j$. Then MRTS of a factor $l$ for factor $l'$ is $\frac{\partial f\_j/\partial y\_{lj}}{\partial f\_j/\partial y\_{l'j}}=\frac{\partial F\_j/\partial y\_{lj}}{\partial F\_j/\partial y\_{l'j}}$, which is MRT of $l$ for $l'$.


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