Question: Game Theory:

Osborne 177.1: (firm-union bargaining): A firm's output from \(L\) units of labor is

\(\left\{\begin{aligned} L(100-L) & \text { if } L \leq 50 \\ 2500 & \text { if } L>50 \end{aligned}\right.\)

the price of output is \(1 . \mathrm{A}\) union representing the workers presents a wage demand \(0 \leq w \leq\) 100, which the firm either accepts or rejects. If the firm accepts the demand, it chooses the number \(L\) of works to employ (take this as continuous, no integer restriction); if it rejects the demand, no production takes place \((L=0)\). The firm cares about profit, while the union wishes to maximize \(wL\).

Find the SPNE of this game. That is:

1. Start with the firm's decision: if it accepts a wage \(w\), what \(L\) should it choose to maximize profits? And which values of \(w\) should it accept? Assume profits are equal to output minus labor cost \(wL\) , and that rejecting \(w\) results in no production, hence zero profits.
2. Given your answer to (a), find the union's optimal demand \(w\).

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