Question:

Given solution:

In the first part of the given solution, can you please explain how and WHY you set 17.3-0.12Qr = 140-10Qr? How did the solution get the "equivalently, Qw = 140-10Qr"?

They can either sell the land as a whole to Steve Win who will build his casino and then once his casino is finished, he will subdivide his land and sell the rest to another casino builder. The alternative is that Atlantic City will itself subdivide the land and sell the plots simultaneously to Steve Win and his rival. Assume that Atlantic City will choose the alternative with the highest industry profits. The estimates that the demand for gambling in Atlantic City (after accounting for the presence of the rest of the casinos in Atlantic City) is P=7505Q where P is the price associated with gambling and Q is the quantity of gambling (think of P as the average amount that a typical patron will net the casino, an amount paid for the entertainment of gambling, and Q as the number of gamblers). Win's total cost of producing gambling is TCw=20+40Qw+15.5Qw2 where Qw is the number of gamblers in Win's casino, and the total cost of producing gambling for Win's rival is TCr=10+50Qr+20Qr2 where Qr is the number of gamblers in the rival's casino and Qw+Qr=Q. Should Atlantic City sell to Steve Win (who will act as a duopolist who moves first) or should it sell to Steve Win and his rival (who will act as cournot or simultaneous duopolists)? a. Suppose they act as a simultaneous duopolist: P=7505Q=7505(Qr+Qw) -For Steve Win: TRw=PQw=(7505(Qr+Qw))Qw=750Qw5Qwr5Qw2 MRw=dTRw/dQw=7505Qr10Qw Since TCw=20+40Qw+15.5Qw2 We have: MCw=40+310w MRw=MCw implies: 7505Qr10Qw=40+31Qw Steve Win's reaction function: Qw=17.30.12Qr -For the rival: TRr=PQr=(7505(Qr+Qw))Qr=750Qw5QwQr5Qr2MRr=dTRr/dQr=75010Qr5Qw Since TCr=10+50Qr+20Qr2 We have: MCr=50+40Qr MRr=MCr implies: 75010Qr5Qw=50+40Qr Rival's reaction function: Qr=140.1Qw. Equivalently, Qw =14010Qr. (2) Setting (1) and (2) equal: 17.30.12Qr=14010Qr Qr=12.42Qw=14010Qr=15.8Q=Qr+Qw=28.2P=7505Q=750528.2=609 Profits - TR-TC 4015.8+15.5+15.82)=5101 For the rival: Profit =TRrTCr=PQr(10+50Qr+20Qr)=60912.42(10+5012.42 +2012.422)=3848 Total profit 5101+38488949 b. Suppose they act as a duopolist where Steve Win moves first: Starting from the rival, its reaction function remains the same: Q=140.1Qw Steve Win's revenue: TRw =PQw=(7505Qw5Qr)Qw=(7505Qw5(14 0.1Qw))Qw=680Qw4.5Qw2 MRw=dTRw/dQw=68090w MRw=MCw implies: 68090w=40+31Qw Qw=16 Qr=140.10w=140.116=12.4 Q=Qw+Qr=16+12.4=28.4 P=7505Q=7505+28.4=608 Profits =TRTC For Steve Win: Profit - TRw-TCw = P*Qw (20+40Qw+15.5Qw2)=60816(20+4016 +15.5162)5100 For the rival: Profit = TRr-TCr=PQr(10+50Qr+20Qr2)=60812.4(10+5012.4+ 2012.42)=3834 Total profit =5100+3834=8934 Atlantic City is betier off selling the plots simultancocaly to Steve Win and his rival because the induatry profits are larger (8049>8934)