Marginal productivity of aggregate labor relation to wage I know that $f'(l)=w$. But if i have that the aggregate production is $Y=AL^{1-α}$  with $L=N*l$ (The number of workers, $N$, multiplied by the labor force). So basically I want to find an expression for $w$ (the wage.).



Full problem:

Consider a one period economy, with aggregate technology given by: $Y=AL^{1-α}$ ,$0&lt;α&lt;1$. $L$ denotes the aggregate labor. The economy is integrated by $N$ identical agents, that offers labor force for a remuneration $w$. Also each person receives an Nth part of the earnings $π$ that generates the aggregate production. The objective of each agent is:

$\max \qquad γ\ln(1-l) + \ln(c)$. 

$\text{subject to} \qquad c = wl + \frac{1}{Nπ(w)}$
So the firm's problem is, letting $L=Nl$

$\max_{L} Y-Lw\implies w=(1-\alpha)AL^{-\alpha}, \pi(w)=\alpha AL^{1-\alpha}$

Consumer's problem: substitute out c and only choose l, taking $\pi$ and $w$ as given

$\max_l u(l, wl+1/(N\pi))\implies -\gamma/(1-l)+w/(wl+1/(N\pi))=0$

Now substitute in the values for w and profit from the firm into the consumer's FOC and you will solve for labor supply (=labor demand as we are in eq) in terms of exogenous parameters ($\gamma, N,\alpha, A)$.