A firm producing two goods; x and y has the profit function given by π=64x-2x^2+4xy-4y^2+32y-14.
What quantities of the two outputs should the firm produce in order to maximise the profit?
To maximize the profit, we need to find the quantities of goods x and y that maximize the profit function π.
To determine these quantities, we need to find the critical points of the profit function. To do this, we take the partial derivatives of the profit function with respect to x and y, and set them equal to zero:
∂π/∂x = 64 - 4x + 4y = 0 (equation 1)
∂π/∂y = 4x - 8y + 32 = 0 (equation 2)
Solving equations 1 and 2 simultaneously will give us the values of x and y that maximize the profit.
From equation 2, we can solve for x in terms of y:
4x = 8y - 32
x = 2y - 8 (equation 3)
Substituting equation 3 into equation 1:
64 - 4(2y - 8) + 4y = 0
64 - 8y + 32 + 4y = 0
36 - 4y = 0
4y = 36
y = 9
Substituting y = 9 into equation 3 to find x:
x = 2(9) - 8
x = 10
Therefore, the quantities of goods x and y that the firm should produce to maximize profit are x = 10 and y = 9.