Given the profit function π = 160x – 3x^3 - 2xy – 2y^2 + 120y – 18 for a firm producing two goods x and y.
i. Find the profit maximizing level of outputs
ii. What is the profit in (i) above?
ii. Test the second order conditions
To find the profit maximizing level of outputs, we need to find the point where the gradient of the profit function is equal to zero.
First, let's find the partial derivatives of the profit function with respect to x and y:
∂π/∂x = 160 - 9x^2 - 2y
∂π/∂y = -2x + 120 - 4y
Setting both partial derivatives to zero, we have:
160 - 9x^2 - 2y = 0 ...(1)
-2x + 120 - 4y = 0 ...(2)
Multiplying equation (2) by 2, we get:
-4x + 240 - 8y = 0 ...(3)
Adding equations (1) and (3), we have:
160 - 9x^2 - 2y - 4x + 240 - 8y = 0
-9x^2 - 4x - 10y + 400 = 0
Rearranging the equation, we get:
9x^2 + 4x + 10y = 400 ...(4)
Now, let's solve equations (2) and (4) simultaneously.
From equation (2):
-2x + 120 - 4y = 0
-2x = 4y - 120
x = -2y + 60 ...(5)
Substituting equation (5) into equation (4):
9(-2y + 60)^2 + 4(-2y + 60) + 10y = 400
9(4y^2 - 240y + 3600) - 8y + 240 + 10y = 400
36y^2 - 2160y + 32400 - 8y + 240 + 10y = 400
36y^2 - 2158y + 32560 = 400
36y^2 - 2158y + 32160 = 0
Now we can solve this quadratic equation to find the value(s) of y.
Using the quadratic formula:
y = (-(-2158) ± √((-2158)^2 - 4*36*32160))/(2*36)
y = (2158 ± √(4657764 - 46080*32160))/72
y = (2158 ± √(4657764 - 1473408000))/72
y = (2158 ± √(-1466750236))/72
It appears that the discriminant is negative, indicating that the quadratic equation has no real solutions. This means that there is no profit maximizing level of outputs.
Therefore, the profit function does not have a maximum point.
ii. Since there is no profit maximizing level of outputs, the profit in part (i) above is not applicable.
ii. The second order conditions cannot be tested, as there is no maximum point to test.