# 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: $Y=AL^{1-α}$ ,$0<α<1$. $L$ 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:

$\max \qquad γ\ln(1-l) + \ln(c)$.

$\text{subject to} \qquad c = wl + \frac{1}{Nπ(w)}$

• What is $f(\cdot)$? – Kun Sep 10 '16 at 1:44
• @Kun The exercise doesn't gives you the production function a single one. Also it doesn't says anything about if the firms are all equal or the number of firms. – neto333 Sep 10 '16 at 1:49
• Do you guys think that something is missing in the exercise? – neto333 Sep 10 '16 at 2:01
• Giving us the entire text of the problem would be helpful. – VCG Sep 10 '16 at 2:13
• Consider a one period economy, with aggregate technology given by: $Y=AL^{1-α}$ ,$0<α<1$. L denotates 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 objetive of each agent is: max $γln(1-l)+ln(c)$. subjet to $c=wl+1/Nπ(w)$ – neto333 Sep 10 '16 at 2:25

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 ($\gamma, N,\alpha, A)$.