Question

Pizza King (PK) and Noble Greek (NG) are competitive pizza chains.  PK believes there is a 30% chance that NG will charge $8 per pizza, a 50% that NG will charge $10 per pizza, and a 20% chance that NG will charge $12 per pizza.  If PK charges price p1 and NG charges price p2, PK will sell 100 + 25(p2 – p1) pizzas. It costs PK $6 to make a pizza. PK is considering charging $7, $8, $9, $10, or $11 per pizza.  To maximize its expected profit, what price should PK charge for a pizza?

Answer 1

1. For price = $7

Number of pizzas sold by PK = (0.3*(100 +25*(8 - 7))) + (0.5*(100 + 25*(10 - 7))) + (0.2*(100 + 25*(12 - 7)))

= 37.5 + 87.5 + 45 = 170

Cost of pizza = $6

Expected profit = 170 * (7 - 6) = $170

2. For price = $8

Number of pizzas sold by PK = (0.3*(100 +25*(8 - 8))) + (0.5*(100 + 25*(10 - 8))) + (0.2*(100 + 25*(12 - 8)))

= 30 + 75 + 100 = 205

Cost of pizza = $6

Expected profit = 205 * (8 - 6) = $410

3. For price = $9

Number of pizzas sold by PK = (0.3*(100 +25*(8 - 9))) + (0.5*(100 + 25*(10 - 9))) + (0.2*(100 + 25*(12 - 9)))

= 25 + 62.5 + 87.5 = 175

Cost of pizza = $6

Expected profit = 175 * (9 - 6) = $525

4. For price = $10

Number of pizzas sold by PK = (0.3*(100 +25*(8 - 10))) + (0.5*(100 + 25*(10 - 10))) + (0.2*(100 + 25*(12 - 10)))

= 16.67 + 50 + 75 = 141

Cost of pizza = $6

Expected profit = 141 * (10 - 6) = $564

5. For price = $11

Number of pizzas sold by PK = (0.3*(100 +25*(8 - 11))) + (0.5*(100 + 25*(10 - 11))) + (0.2*(100 + 25*(12 - 11)))

= 8.33 + 37.5 + 62.5 = 108

Cost of pizza = $6

Expected profit = 108 * (11 - 6) = $540

Hence in order to maximise its expected profit, PK should charge $10 per pizza since expected profit is maximum, i.e. $564 in this case.