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I dont want to be rude but the only equation you copied correctly is the productivity augmented Cobb-Douglas production function. Equation 2 is equation 2 from Ottaviano, Peri (2008) (on page 8) it says: $$L=\left(\sum_{e=1}^{n} \theta_{e}L_{e}^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$. In the remaining equations $\theta$ is not under ...
As alluded to by Bertrand in his +1 comments this is because FOCs do not tell you where maximum or minimum occurs. This is common misconception among some students but it simply does not hold. FOCs give you location of stationary points. They give you points where $df(x,y)/dx = df(x,y)/dy= 0$. A maximum or minimum will occur at points where \$df(x,y)/dx = df(...