Suppose that Alpha AS and Beta AS manufacture competitive products, with the weekly sales of each product determined by the selling price of that product and the price of its competition. Suppose that Alpha sets a sales price of x dollars per unit for its product, while Beta sets a sales price og y dollars per unit for its product. Market research shows that the weekly profit made by Alpha is then
P(x) = -2x^2 + 12x + xy - y - 10
and that the weekly profit made by Beta is
Q(y) = -3y^2 + 18y +2xy -2x - 15
(both in thousands of dollars). The peculiar notation arises from the fact that x is the only variable under the control of Alpha and y is the only variable under the control of Beta.
Assume that both company managers know calculus and that each knows that the other knows calculus and has some common sense. What price will each manager set to maximize his company's weekly profit?
The Attempt at a Solution
I find the partial derivates dP/dx and dQ/dy, make them equal 0 and find x and y from the two equations y = 4x - 12 and x = 3y - 9.
x = 0,82 dollars
y = 3,27 dollars
However, the correct answer is supposed to be
x = 4,09 dollars
y = 4,36 dollars
What's my mistake?