Question

Jane and Sara are competing orange juice salespersons in Amherst. Their stands are next to each other on a street and consumers regard them as identical. The marginal cost of an orange juice is $1. The demand for orange juice every hour is Q = 20 − P where P is the lowest price between the two salespersons. If their prices are equal they split demand equally.

a. If they set prices simultaneously (prices can be any real number), what is the Nash equilibrium price?

b. If, against what we have assumed in class, orange juice salespersons have to charge prices in whole dollars ($1, $2, $3, etc), what are the Nash equilibrium prices?

c. Assuming whole dollar pricing, if Jane sets her price before Sara, what price would she charge?

Answer all 3 parts if you dont mind.

Answer 1

The Nash equilibrium price is when Jane and Sara both set the same price. If orange juice salespersons have to charge prices in whole dollars, the possible Nash equilibrium prices are 1, 2, 3, etc. If Jane sets her price before Sara, Jane would charge $1.

Step-by-step explanation:

a. To find the Nash equilibrium price when Jane and Sara set prices simultaneously, we need to analyze their demand functions. The demand function for orange juice is given by Q = 20 - P. Since the prices are the same for both salespersons, they split the demand equally, so each person gets half of the demand.

To find the Nash equilibrium, we need to set up a system of equations. Let x be the price set by Jane, and y be the price set by Sara. Therefore, the quantity demanded by Jane is Qj = 20 - x, and the quantity demanded by Sara is Qs = 20 - y. Since they split the demand equally, Qj = Qs. Substituting values, we get 20 - x = 20 - y, which simplifies to x = y.

Therefore, the Nash equilibrium price is when Jane and Sara both set the same price.

b. If orange juice salespersons have to charge prices in whole dollars, the possible Nash equilibrium prices are 1, 2, 3, etc.

c. If Jane sets her price before Sara, Jane would charge $1.