I am following (and trying to understand) a paper where the wage of unskilled workers is determined as the outcome of wage bargaining between a single union and a single firm in a right-to-manage framework. There is a single union that represents only the unskilled, and which has a utilitarian utility function of the form:
\begin{equation} V=\frac{1}{\bar{L}} \left[ LU(\tilde{w_{u}}) +(\bar{L}-L)U(B)\right] \end{equation}
where $\bar{L}$ is the unskilled labour force, $U(\cdot)$ is the workers’ utility function, B is the unemployment benefit, and the net wage is given by $\tilde{w_{u}}=(1-\tau) w_{u}$. Workers are further assumed to be risk-averse with utility $U(\tilde{w_{i}})=\tilde{w_{i}}^{\rho}$. The paper than says that wage bargaining process is then governed by (pag. 7): \begin{equation} \begin{matrix} max\\ w_{u} \end{matrix}=([((1-\tau)w_{u})^{\rho}-B^{\rho}])^{\gamma}(Y-w_{u}L-w_{s}H)^{1-\gamma} \end{equation} I simply do not understand this last passage. Shouldn't it be something like this by substituting? Where does the multiplication in the paper come from?
\begin{equation} V=\frac{1}{\bar{L}} \left[ L((1-\tau) w_{u})^{\rho} +(\bar{L}-L)B^{\rho}\right] \end{equation}
\begin{equation} V=\frac{L}{\bar{L}} ((1-\tau) w_{u})^{\rho} + B^{\rho} - \frac{L}{\bar{L}} B^{\rho} \end{equation}
\begin{equation} V=\frac{L}{\bar{L}} \left[((1-\tau) w_{u})^{\rho} -B^{\rho}\right]+ B^{\rho} \end{equation}
I attach an open version of the paper in question and thank anyone who wants to take a look at it and explain. https://air.unimi.it/retrieve/handle/2434/142829/122345/2.pdf
