EXERCISE $177\mathrm{.}1($Firm$-$union bargaining$)$ A firm's output is L$(100-$L$)$ when it uses L units of labor, and $2500$ when it uses L $>50$ units of labor. The price of output is $1.$ A union that represents workers presents a wage demand $($a nonnegative number w $),$ which the firm either accepts or rejects. If the firm accepts the demand, it chooses the number L of workers to employ $($which you should take to be a continuous variable, not an integer$)$; if it rejects the demand, no production takes place $($L$=0).$ The firm's preferences are represented by its profit; the union's preferences are represented by the value of wL$.$

a$.$ Formulate this situation as an extensive game with perfect information.

b$.$ Find the subgame perfect equilibrium $($equilibria$?)$ of the game.

c$.$ Is there an outcome of the game that both parties prefer to any subgame perfect equilibrium outcome?

d$.$ Find a Nash equilibrium for which the outcome differs from any subgame perfect equilibrium outcome.