The two largest diner chains in Kansas compete for weekday breakfast customers. The two chains, Golden Inn and Village Diner, each off weekday breakfast customers a "breakfast club" membership that entitles customers to breakfast buffet between 6AM and 8:30AM Club members are sold as passes good for 20 weekday breakfast visits.
Golden Inn offers a modest but tastey buffet, while village diner provides a wider variety of breakfast items that are also said to be quite tasty. The demand functions for the breakfast club memberships are:
QG = 5,000 - 25PG + 10PV
QV= 4,200 - 24PV + 15PG
Where QG and QV are the number of club memberships sold monthly and PG and PV 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
LACG = LMCG = $50 per membership
LACV = LMCV = $75 per membership
The best response curves for Golden Inn and Village Diner are, respectively,
PG = BRG(PV) = 125 + 0.2PV
PV = BRV(PG) = 125 + 0.3125PG
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?