Consider the following scenario. There is an employer (E), and an employee (U). Following a recent pandemic and the ensuing economic downturn, E and U are negotiating a new wage package. If E and U work together they generate a total value / profit of for the E. Thus the players in the game are E and U who are using alternating offer bargaining to negotiate. There is a third entity - the government (G) who is not a player in the game. However, because of the pandemic emergency, the government, G has promised a remuneration r to the employee U, should the negotiation break down and U is out of job. There are two periods in the game i.e., t = 0, 1. The game proceeds as follows:

in period 0, i.e., at t = 0, the employee U makes a wage offer w to the employer E. The game then moves to the second stage of t = 0, where E has to respond by either accepting or rejecting the offer. If E accepts the offer, the game ends, and the pay-offs are w for U and w for E. If E rejects the offer the game moves to period t = 1.

at t = 1, it is the employer i.e., E's turn to make a counter offer. E makes a wage offer (counter offer ) w' to U. The game then moves to the second stage of period t = 1, where U has to respond by accepting or rejecting the offer. If U accepts E's offer at this stage, the game ends and E's pay off (present-value) is ( w' ), while U's pay-off is w' . The term < 0 < 1, is the common discount factor. If U rejects E's counter-offer the game ends, and U accepts the government's remuneration package r, so U's pay-off in this case is r. The pay-off of E is 0.

In the above 2-period alternating offer Bargaining game, compute the subgame perfect equilibrium (SPE). ( 20 points)