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Could someone please show how the author derives the the first order condition (14) of this optimization problem using the expressions shown here ?
For context this is taken from a labour market model where the private sector wage, w_p , is the result of Nash bargaining over the respective surpluses of the households V^h and firms V^f
Taking the derivative of the goal function w.r.t. $w_t^P$ you get $$ (1- \vartheta)\frac{1}{V^h_{nPt}}\frac{\partial V^h_{nPt}}{\partial w_t^P} + \vartheta\frac{1}{V^f_{nPt}}\frac{\partial V^f_{nPt}}{\partial w_t^P} = 0 $$ which, after substituting the definitions is $$ (1- \vartheta)\frac{1}{V^h_{nPt}}\left(\lambda_{ct}(1-\tau_n^t)\right) + \vartheta\frac{1}{V^f_{nPt}}(-1) = 0. $$ Rearranging $$ (1- \vartheta)\frac{1}{V^h_{nPt}}\lambda_{ct}(1-\tau_n^t) = \vartheta\frac{1}{V^f_{nPt}}, $$ rearranging $$ (1- \vartheta)\lambda_{ct}(1-\tau_n^t) V^f_{nPt} = \vartheta V^h_{nPt}, $$ QED.
