# Partial differentiation (maximize

1. Feb 17, 2007

### kasse

1. The problem statement, all variables and given/known data

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?

3. 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.

This gives

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?

Last edited: Feb 17, 2007
2. Feb 17, 2007

### suspenc3

Im not sure how you found those eqns...I did it and got the correct answers..Your method is correct

3. Feb 17, 2007

### suspenc3

Show me the partials you got for these:

$$P(x) = -2x^2 + 12x + xy - y - 10$$

$$Q(y) = -3y^2 + 18y +2xy -2x - 15$$

4. Feb 17, 2007

### quaker

He shows the partials in the first post, y = 4x-12 and x = 3y-9

5. Feb 17, 2007

### suspenc3

Ohh...Yeah..you must have just made an error calculating X and Y

6. Feb 17, 2007

### kasse

You're right. Damn irritating mistake :grumpy:

How about if the managers entered into an agreement by which they plan to maximize their total weekly profit? What should be the selling price of each product?

7. Feb 17, 2007

### suspenc3

Im kinda confused about your question, Isnt that what you just found?

8. Feb 17, 2007

### kasse

No. It's the part (b) of the task:

"Now suppose that the two managers enter into an agreement (legal or otherwise) by which they plan to maximize their TOTAL weekly profit. Now what should be the selling price of each product? (We suppose that they will divide the resulting profit in an equitable way, but the details of this intriguing problem are not the issue). "

What I tried:

R(x,y) = P(x)+Q(y)
Find the partials, find eq.s for x and y and solve them. This gives y=8,4, but is supposed to be 6,53. I don't think this is the same mistake as I did in part (a).

Last edited: Feb 17, 2007
9. Feb 17, 2007

### kasse

Is this the correct method?