Consider a firm and a temporary worker negotiating the division of the profit. The firm and the worker are in a relationship for two periods only. In each period the worker works, a profit of $1$ is generated. In the first period, the firm makes a demand x$.$ If the worker accepts, this is how the profits are divided: the firm receives $2$x $($x per period$)$ and the worker receives $2(1$ x$).$ If the worker rejects the offer, the firm receives u$1$ and the worker receives u$2,$ and then move to the second period. In the second period, the worker demands y$.$ If the firm rejects, they receive u$1$ and u$2$ respectively. If the firm accepts, the worker receives y of the second period profit and the firm receives $1$y of the second period profit. This game is represented by the following tree. In this exercise, u$1$ and u$2$ are the firm$$s and the worker$$s outside options while the worker is on strike.

firm

x

worker

Reject worker

y

firm

Accept

$2$x $2(1$x$)$

Reject $2$u$1$

Accept u$1+1$ y

u$2+$ y

$($a$)$ Find the subgame Perfect Nash equilibrium $($SPNE$)$ of this game.

$2$

$2$u$2$

$($b$)$Does the equilibrium payoff of the firm increase with u$1$ and$/$or u$2?$

$($c$)$Does the equilibrium payoff of the worker increase with u$1$ and$/$or u$2?$

$($d$)$Suppose that before the game is played, the firm can choose to change its production location in a way that lowers the outside options of both players, that is$,$ lowers both u$1$ and u$2$ to v$1<$ u$1$ and v$2<$ u$2.$ Would the firm ever do that? If yes, give an example of v$1$ and v$2$ for which the firm wants to do that. If not, explain why not.