The two largest diner chains in Kansas compete for weekday breakfast customers. The two chains, Golden Inn and Village Diner, each offer weekday breakfast customers a "breakfast club" membership that entitles customers to a breakfast buffet between 6:00 A.M. and 8:30 A.M. Club memberships are sold as "passes" good for 20 weekday breakfast visits. Golden Inn offers a modest but tasty buffet, while Village Diner provides a wider variety of breakfast items that are also said to be quite tasty. The demand functions for breakfast club memberships are

Q_G = 5,000 - 25P_G + 10P_V

Q_V = 4,200 - 24P_V + 15P_G

where Q_G and Q_V are the number of club memberships sold monthly and P_G and P_V are the prices of club memberships, both respectively, at Golden Inn and Village Diner chains. Both diners experience long-run constant costs of production, which are

LAC_G = LMC_G $50 per membership

LAC_V = LMC_V = $75 per membership

The best-response curves for Golden Inn and Village Diner are, respectively,

P_G = BR_G(P_V) = 125 + 0.2P_V

P_V = BR_V(P_G) = 125 + 0.3125P_G

a.If Village Diner charges $200 for its breakfast club membership, find the demand, inverse demand, and marginal revenue functions for Golden Inn. What is the profit maximizing price for Golden Inn given Village Diner charges a price of $200? Verify mathematically that this price can be obtained from the appropriate best-response curve given above.

b.Find the Nash equilibrium prices for the two diners. How many breakfast club memberships will each diner sell in Nash equilibrium? How much profit will each diner make?

c.How much profit would Golden Inn and Village Diner earn if they charged prices of $165 and $180, respectively? Compare these profits to the profits in Nash equilibrium (part c). Why would you not expect the managers of Golden Inn and Village Diner to choose prices of $165 and $180, respectively?