Optimization problem - wage determination

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

• Why are people so reluctant to name "the author" or "the paper"? Providing a direct link could also be useful! Feb 9, 2022 at 13:48

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.