Net product marginal in Acemoglu's article "Modeling inefficient institutions"

I'm reading lecture notes by Acemoglu regarding institutions and I'm trying to study his famous model of non-democratic institutions, deeply described in his article "Modeling inefficient institutions" Modeling inefficient institutions but I'm struggling with the "net marginal product of a worker employed by a producer of group j. Maximization problem seem easy and I obtain $$k_t^j$$ but when I try to compute $$l_t^j$$ I don't get how to obtain $$\displaystyle \frac{\alpha(1-\tau_t^j)^\frac{1}{\alpha}}{(1-\alpha)}A^j$$. I tried to derive respect o labor the production function (equation (2) in the article) but I don't obtain the same result, even after a multiplication with taxes). Anyone can explain how to obtain this expression? I'm thinking that is very simple but I really don't get what I'm not taking account of.

Calculate the first-order-condition (FOC) wrt labour: \begin{align}\frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha - w_t l_t^j - k_t^j \right) &= 0 \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{(k_t^j)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &= w_t \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{\left((1-\tau_t^j)^{\frac{1}{\alpha}} A^j l_t^j\right)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &= w_t \\ \implies \frac{\alpha}{1-\alpha} (1-\tau_t^j)^{1/\alpha} A^j &= w_t \end{align}
While calculating the FOC, substitute $$k_t^j$$ with the optimal value (already) obtained from the other FOC. This is marked in red above.
net marginal product of a worker employed by a producer of group $$j$$
In case you don't know what that means, it is the partial derivative of the output $$(y_t^j)$$ wrt labour $$(l_t^j)$$. In other words, it is $$\frac{\partial y_t^j (l_t^j, k_t^j)}{\partial l_t^j} = \frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha \right)$$.
• Thank you very much. I was expecting to find an expression like $l_t^j=...$ and the real answer was given by the equality between wage and the FOC respect to labor given the solution of the problem for $k_t^j$. Yes I know the meaning of "net..." but computing the derivative I was expecting the same expression of the article without substitution of the optimal value for $k_t^j$. Your answer was very clarifying Mar 31 at 8:22