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


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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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