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
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 $l'$.