I can`t quite wrap my head around one step in deriving the pigou tax in a multiple polluter case.

There are j firms, which produce joint product in amount of $k_j$. Joint product can be abated, where the cost function $C_j(a_j)$ denotes the costs for firm j for abating $a_j$

Social damage costs: $ e = \sum_{j=1}^{m}{(k_j -a_j) }$

To minimize total social cost:

$min_{(aj)_{j=1}^{m}}\sum_{j=1}^{m}C_j(a_j)+ \sum_{i=1}^{m}D_i(e)$

With FOC we should, according to my script, arrive at:

$C_j^`(a_j)=\sum_{i=1}^{n}D_j^`(e)$ for all j = 1,...,m

But I don't understand why it isn't: $\sum_{j=1}^{m}C_j^`(a_j)=-\sum_{i=1}^{n}D_j^`(e)$

I think for example the partial derivative of $\sum_{j=1}^{2}C_j$ is equal to the partial derivative of it's parts. So $[\sum_{j=1}^{2}C_j]' = C_{1}^{'}(a_1) + C_{2}^{'}(a_2)$ and therefore:

$\sum_{j=1}^{m}C_j^`(a_j)+\sum_{i=1}^{n}D_j^`(e)= 0$

I don't really know, where I`m going wrong with this and would be really happy if someone could point me into the right direction.

Best regards :)



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