# 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 $=(L(100-L) - wL) \quad if \quad L<=50 \quad or\quad (2500 - wL) \quad if \quad L>50$

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?

## 2 Answers

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 $v_F+v_U$.

• I am agree with @Michael Greinecker ,you just have to do this max $Max(w,L) \quad L(100−L)−wL) \quad s.t. \quad U_o = wL \quad if \quad L<=50$ and $Max(w,L) \quad 2500-wL \quad s.t.\quad U_o = wL \quad if \quad L>50$ do both lagrangians and then you will find the Pareto optimum, maybe is a set !. I put the equalities cause Walras theorem, it will not affect the optimum in this case. Remember $U_o$ is a constant for the Lagrangian problem! May 7 '18 at 5:13

To find the Pareto optimal outcomes, simply maximize the total income of the labor and the firm. \begin{eqnarray*} \max_{L} \ \ \begin{cases} L(100-L) & \text{if } L \leq 50 \\ 2500 & \text{if } L > 50 \end{cases}\end{eqnarray*} Solution to the above problem is any $L \geq 50$. The corresponding maximum total income is 2500, which can now be divided among the labor and the firm in any way, thus yielding all Pareto optimal allocations. Therefore, set of all Pareto optimal outcomes where both the firm and the labor get non-negative payoffs is : $\{(w, L) \in \mathbb{R}^2_+ : L \geq 50, wL \leq 2500 \}$

• Dear Amit please can you look at my question? I always see subgame perfect equilibrium questions. But I don’t understand how to solve. I post a question as an example of SPE. Please show it. Thanks a lot. economics.stackexchange.com/questions/21879/…
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May 8 '18 at 2:36