Reference no: EM133081090
Suppose that there are two food stores in town. La Boulangerie sells bread and La Fromagerie sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If La Boulangerie's price is PB dollars per loaf of bread and La Fromagerie's price is PF dollars per pound of cheese, their respective weekly sales, QB thousand loaves of bread and QF thousand pounds of cheese, are given by the following equations:
QB = 14 - PB - 0.5PF
QF = 19 - 0.5PB - PF
a.) What are profit functions for La Boulangerie, ΠB, and La Fromagerie, ΠF?
b.) What are the best response functions in terms of setting the prices for La Boulangerie and La Fromagerie?
c.) Graph the respective best-response functions. Plot La Boulangerie's price on the vertical axis and La Fromagerie's price on the horizontal axis.
d.) Calculate the Nash equilibrium
Rather than setting their prices independently, suppose now that the two stores collude to set prices jointly so as to maximize the sum of their profits. Over the next few questions, we are going to find the joint profit maximizing prices for the stores.
e.) Start by identifying the profit function that captures the sum of the profits for the two stores i.e. ΠJ = ΠB + ΠF.
f.) What price should the two stores set for La Boulangerie? In other words, what PB maximizes the joint profit function? This looks a little trickier because the joint profit function, if you've calculated it correctly, has two squared terms in it instead of one. Think about it this way, though. Recall that the generic quadratic equation looks like y = ax2 + bx + c. In this part of the question, the x we're interested in is PB. Any term in our joint profit function that doesn't have some kind of PB in it is just part of the constant, 'c'. The 'x' that maximizes 'y' is still -b/2a. With this in mind, what is the best response function for the price set by La Boulangerie, PB?
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