Little Italy has 300 consumers all of whom buy either one pizza or nothing. Each consumer has a reservation price of $14. That is, each consumer is willing to pay at most $14 for a pizza. There are two pizza shops - A and B - located next to each other. It costs $7 to make a pizza. Let Pa and Pb denote the price (in dollars) charged by shop A and shop B respectively. For simplicity, assume that prices lie between 8 and 14 (including these two prices). That is, for i=A, B. Pi element of {8,9,10,11,12,13,14}.
Assumption A: if Pj-1 <= Pi <= Pj+1, half of the consumers buy from shop i while the other buy from shop j, where i,j element of {A, B} and i not equal to j. Else if Pi <= Pj-2, all consumers buy from shop i, while if Pi >= Pj+2, all consumers buy from shop j. For example, if PB=11, consumers are equally split between A and B as long as 10 <= PA <= 12. Else, if PA <= 9, all consumers buy from shop A, while if PA >= 13, all consumers buy from shop B.
Let Qi denote the number of consumers that buy from shop i. Then, shop i's profit is Phi_i = (Pi-7)Qi. Each shop maximizes its own profit.
Suppose Pb = 13q + 10(1-q) where 0 <= q <= 1.