Let $F_j$ denote the transformation function of firm $j$, i.e., $j$'s production set $Y_j=\{y:F_j(y)\le0\}$. Also, let $F$ denote the transformation function of the market, i.e., $Y=\{\sum_j y_j: y_j\in Y_j\}=\{y:F(y)\le0\}$. Is there any relationship between $F_j$'s and $F$?

The motivation behind this question is the following statement regarding MRTs:

$\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}=\frac{\partial F_{j'}/\partial y_{lj'}}{\partial F_{j'}/\partial y_{l'j'}}$ for all $j,j',l,l'$ implies $\frac{\partial F_1/\partial y_{l1}}{\partial F_1/\partial y_{l'1}}=\frac{\partial F/\partial y_{l}}{\partial F/\partial y_{l'}}$ for all $l,l'$

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