How to calculate Pareto Optimal outcome in a game with a Nash Equilibrium I have at hand a two period game. 

In period 1, Union decides wage offer $w$. 

In period 2, firm decides on L, given wage. 

Union maximizes $wL$

Firm maximizes $=(L(100-L) - wL) \quad if \quad L&lt;=50 \quad or\quad (2500 - wL) \quad if \quad L&gt;50 $

An SPNE of this game is: w = 50, L = 25. 

I need to compute a PO allocation. I know it exsits, but I'm unsure about how to derive it from the payoffs. (Example: w = 40, L = 40. Gives both firm and union strickly greater payoffs.) But, how do I compute the set of PO outcomes from the gives functions? Or even a single PO outcome, without guesswork?
For Pareto optimality, you can ignore the timing. Also, anything you know about Nash equilibria in this game is irrelevant.

Let $v_F(w, L)$ be the payoff of the firm and $v_U(w, L)$ be the payoff of the union given $(w, L)$. What you are looking for are pairs $(w, L)$ such that neither the firm nor the union can have a higher payoff without the other side having a lower payoff. Equivalently, you want to the payoff of the firm to be as large as possible under the constraint that the union cannot have a lower payoff and you want the payoff of the union to be as large as possible under the constraint that the firm cannot have a lower payoff. 

This means that $(w,L)$ is Pareto optimal if and only if it solves the following two maximization problems:

$$\max_{(w',L')} v_F~\text{ s.t.}~v_U(w',L')\geq v_U(w,L).$$
$$\max_{(w',L')} v_U~\text{ s.t.}~v_F(w',L')\geq v_F(w,L).$$

Now if you just have to find one Pareto optimum and not all Pareto optima, you can just maximize $v_F+v_U$.