The two largest diner chains in Kansas compete for weekday breakfast consumers. The two chains, Golden Inn and Village Diner, each offer weekday breakfast customer a �breakfast club� membership that entitles customers to a breakfast buffet between 6:00 a.m. to 8:30 a.m. Club membership 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:
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 costs of production, which are
LACG = LMCG = $ 50 per membership
LACV = LMCV = $ 75 per membership
The best response curves for Golden Inn and Village Diners 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?
Please type your subject in the School Subject box. Any other words, including obscure abbreviations, are likely to delay responses from a teacher who knows that subject well.
Also, please understand that no one here will do your work for you. However, we will be happy to read over whatever you come up with and make suggestions and/or corrections.
Please post what you think.
a. To find the demand function for Golden Inn (QG), we substitute the given price for Village Diner (PV) into the demand function equation:
QG = 5,000 - 25PG + 10PV
QG = 5,000 - 25PG + 10(200)
QG = 5,000 - 25PG + 2,000
QG = 7,000 - 25PG
The inverse demand function for Golden Inn (PG) can be obtained by rearranging the demand function:
PG = (7,000 - QG) / 25
The marginal revenue (MR) function for Golden Inn is the derivative of the inverse demand function with respect to quantity:
MRG = d(PG) / dQG = -1/25
To find the profit-maximizing price for Golden Inn given Village Diner charges $200, we set the marginal cost (LMCG) equal to the marginal revenue (MRG):
LMCG = MRG
$50 = -1/25
Solving the equation, we get:
$50 = -1/25
$50 = -0.04
Since this equation has no solution, it means that there is no profit-maximizing price for Golden Inn given that Village Diner charges $200. This is because the cost is higher than the revenue.
To verify mathematically that this price can be obtained from the best response curve, we substitute the given price for Village Diner (PV) into the best response curve equation for Golden Inn (PG):
PG = 125 + 0.2PV
PG = 125 + 0.2(200)
PG = 125 + 40
PG = 165
Therefore, the price of $165 can be obtained from the best response curve.
b. To find the Nash equilibrium prices for the two diners, we need to find the intersection point of their best response curves. We set the two best response curves equal to each other and solve for the prices (PG and PV):
PG = 125 + 0.2PV
PV = 125 + 0.3125PG
By substituting the first equation into the second equation, we get:
PV = 125 + 0.3125(125 + 0.2PV)
PV = 125 + 39.0625 + 0.0625PV
0.9375PV = 164.0625
PV = 164.0625 / 0.9375
PV = 175
Substituting this result back into the first equation, we find:
PG = 125 + 0.2(175)
PG = 125 + 35
PG = 160
Therefore, the Nash equilibrium prices for Golden Inn and Village Diner are $160 and $175, respectively.
To calculate the number of breakfast club memberships sold, we substitute these prices back into their respective demand functions:
QG = 7,000 - 25PG
QG = 7,000 - 25(160)
QG = 7,000 - 4,000
QG = 3,000
QV = 4,200 - 24PV
QV = 4,200 - 24(175)
QV = 4,200 - 4,200
QV = 0
In Nash equilibrium, Golden Inn would sell 3,000 breakfast club memberships, while Village Diner would not sell any.
To calculate the profit for each diner, we multiply the quantity sold by the price and subtract the total cost:
ProfitG = (PG * QG) - (LMCG * QG)
ProfitG = (160 * 3,000) - (50 * 3,000)
ProfitG = 480,000 - 150,000
ProfitG = 330,000
ProfitV = (PV * QV) - (LMCV * QV)
ProfitV = (175 * 0) - (75 * 0)
ProfitV = 0 - 0
ProfitV = 0
Therefore, Golden Inn would make a profit of $330,000, while Village Diner would make a profit of $0.
c. If Golden Inn charged a price of $165 and Village Diner charged a price of $180, we can calculate the profits for each diner using the same process:
ProfitG = (PG * QG) - (LMCG * QG)
ProfitG = (165 * QG) - (50 * QG)
ProfitG = 115 * QG
ProfitV = (PV * QV) - (LMCV * QV)
ProfitV = (180 * QV) - (75 * QV)
ProfitV = 105 * QV
Since we don't know the exact quantities sold at these prices, we cannot calculate the exact profits. However, both diners would likely make less profit than in the Nash equilibrium.
Managers of Golden Inn and Village Diner would not choose prices of $165 and $180, respectively, because these prices would not maximize their profits. In Nash equilibrium, the prices are set at levels where both diners can maximize their profits. Charging higher or lower prices than the Nash equilibrium would result in lower profits or no sales for one or both diners. Therefore, the managers would choose the Nash equilibrium prices to maximize their profits.